The stiff Neumann problem: Asymptotic specialty and “kissing” domains

Abstract

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain $Ω \subset R^{d}$ which is divided into two subdomains: an annulus $Ω_{1}$ and a core $Ω_{0}$ . The density and the stiffness constants are of order $ε^{- 2 m}$ and $ε^{- 1}$ in $Ω_{0}$ , while they are of order 1 in $Ω_{1}$ . Here $m \in R$ is fixed and $ε > 0$ is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as $ε \to 0$ for any m. In dimension 2 the case when $Ω_{0}$ touches the exterior boundary $\partial Ω$ and $Ω_{1}$ gets two cusps at a point $O$ is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as $x \to O$ for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

Keywords

Neumann Laplacian asymptotics of eigenvalues and eigenfunctions stiff Neumann problem domain with cuspidal point kissing domains

1. Introduction

Let Ω be a smooth bounded domain in $R^{d}$ and let $Ω_{1}$ and $Ω_{0}$ be two bounded domains in $R^{d}$ with smooth boundaries $Γ_{1}$ and $Γ_{0}$ respectively such that $\partial Ω = Γ_{1}$ , ${\overline{Ω}}_{0} \subset Ω$ and $Ω = Ω_{0} \cup Ω_{1} \cup Γ_{0}$ . We refer to $Ω_{1}$ as the annulus and $Ω_{0}$ as the core. A typical geometrical situation is drawn in Fig. 1, where the annulus is shaded. We consider the spectral Neumann problem in $Ω_{1} \cup Ω_{0}$ with natural transmission conditions for a second order differential operator with piecewise constant coefficients $\begin{array}{l} (1.1) & - Δ_{x} u_{1}^{ε} (x) = λ^{ε} u_{1}^{ε} (x), x \in Ω_{1}, \\ (1.2) & - ε^{- 1} Δ_{x} u_{0}^{ε} (x) = λ^{ε} ε^{- 2 m} u_{0}^{ε} (x), x \in Ω_{0}, \\ (1.3) & \partial_{ν_{1}} u_{1}^{ε} (x) = 0, x \in Γ_{1}, \\ (1.4) & u_{0}^{ε} (x) = u_{1}^{ε} (x), ε^{- 1} \partial_{ν_{0}} u_{0}^{ε} (x) = \partial_{ν_{0}} u_{1}^{ε} (x), x \in Γ_{0}, \end{array}$ where $\partial_{ν_{1}}$ and $\partial_{ν_{0}}$ denote the derivatives along outward and inward normal vectors $ν_{1}$ and $ν_{0}$ to $Γ_{1}$ and $Γ_{0}$ respectively, $λ^{ε}$ is the spectral parameter and $- 2 m \in R$ a fixed exponent. From a physical point of view, the factor $ε^{- 2 m}$ reflects the dead-weight of the material, i.e. increasing m makes the material heavier.

Fig. 1.

Annulus domain $Ω_{1}$ and core domain $Ω_{0}$ .

The aim of this paper is the description of the asymptotic behaviour as $ε \to 0$ of the eigenpairs $(λ^{ε}, u^{ε})$ of spectral problem (1.1)–(1.4), where the (real) eigenfunctions $u^{ε}$ are identified with the pairs of functions ${u_{0}^{ε}, u_{1}^{ε}}$ with $u_{i}^{ε}$ the restriction of $u^{ε}$ to $Ω_{i}$ , for $i = 0, 1$ .

According to the range in which the parameter $m \in R$ varies, different ansätze are needed to describe the asymptotic behaviour of the eigenpairs $(λ^{ε}, u^{ε})$ , as $ε \to 0$ (see Section 3 for the case $m \in (0, 1 / 2)$ and Sections 4–7 for the other values of m). The main result of this paper is stated in Theorem 2.1 which provides the error estimates of $(λ^{ε}, u^{ε})$ and holds for any value of $m \in R$ . We detail the proof only in the case $m \in (0, 1 / 2)$ where a re-normalization of the eigenfunctions is the key point (cf. formula (3.1)). The proof of Theorem 2.1 is split into two steps. The first step is a convergence result (see Proposition 3.1) and it shows that, in the case $m \in (0, 1 / 2)$ , the eigenvalue $λ_{n}^{ε}$ of problem (1.1)–(1.4) converges to some eigenvalue $λ_{\bar{n}}^{0}$ of the limit problem $\begin{array}{l} - Δ_{x} v_{1}^{0} (x) = λ^{0} v_{1}^{0} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} v_{1}^{0} (x) = 0, x \in Γ_{1}, v_{1}^{0} (x) = 0, x \in Γ_{0} . \end{array}$ The second step consists to proving that $n = \overline{n}$ . To this end, the key ingredient is the so-called “Lemma about near eigenvalues and eigenfunctions” (cf. [24]), which, in the case $m \in (0, 1 / 2)$ , requires a non-standard choice of the approximate eigenfunctions and the use of the Neumann series in order to represent the solution of an auxiliary boundary value problem (see Lemma 3.3). For the other values of m, the proof of Theorem 2.1 follows the same strategy as in the case $m \in (0, 1 / 2)$ but it requires no special tools and the choice of the almost eigenvalues and almost eigenfunctions (also called quasimode) are quite standard. However, for the sake of completeness, we provide all the necessary changes to prove Theorem 2.1 (see Sections 4–7). We also stress out that the case $m = 1 / 2$ has been previously investigated in [23, Chapter VII] in a different setting. However, we give an independent proof of the justification of the anzätze (see Section 6).

The spectral problems (1.1)–(1.4) are of interest in many area of physics, such as the study of reinforcement and elasticity problems (cf. [1–3,21]). In [13], estimates of convergence rates of the spectrum of stiff elasticity problems have been obtained. We also mention the papers [6,8], where the authors have dealt with the asymptotics of spectral stiff problem in domains surrounded by a thin band depending on ε. For a study of asymptotics for vibrating systems containing a stiff region independent of the small parameter ε, we refer to Sections V.7–V.10 in [23] and the papers [7,12,22].

In the last part of the paper, the spectral problem (1.1)–(1.4) is discussed when the domain Ω becomes irregular (see Section 8). More specifically, a irregular point appears on the boundary $\partial Ω$ , consisting of the point $O$ of tangency of the two “kissing” disks $Ω_{0}$ and $Ω_{1}$ in $R^{2}$ (cf. Fig. 2). In the case of “kissing” domains, the perturbation analysis previously performed could not repeat because of possible singularities of $u_{0}^{ε}$ and $u_{1}^{ε}$ of problem (1.1)–(1.4) at the irregular point $O$ (cf. Section 8.4). We show that these singularities do not affect our asymptotic procedure in the stiff Neumann problem (1.1)–(1.4) so that the ansätze obtained when the boundary $\partial Ω = Γ_{1}$ is smooth are still valid. It is worth to mention that in a certain sense the problem (1.1)–(1.4) can be reduced to a regular perturbation in an operator setting depending on the exponent m. In this way, the full asymptotic series for eigenpairs of the problem can be readily derived in the “smooth” case after constructing the main asymptotic and first correction terms. On the contrary, if we consider a stiff Dirichlet problem, namely when the condition (1.3) is replaced by $u_{1}^{ε} (x) = 0$ , $x \in Γ_{1}$ , the asymptotic procedure requires important changes (see Section 8.4) and further investigations of a stiff Dirichlet problem are left as open questions.

For $m ⩽ 1 / 2$ , the limit problem in the cuspidal annulus $Ω_{1}$ is given by $\begin{array}{l} (1.5) & - Δ_{x} u (x) = λ u (x), x \in Ω_{1}, \\ (1.6) & \partial_{ν_{1}} u (x) = 0, x \in Γ_{1} ∖ O, u (x) = g (x), x \in Γ_{0} ∖ O, \end{array}$ where $λ ⩾ 0$ and $g = 0$ or $g = const$ on the boundary $Γ_{0}$ . Thanks to the Dirichlet boundary condition on $Γ_{0}$ , $H_{0}^{1} (Ω_{1}; Γ_{0})$ is compactly embedded into $L^{2} (Ω_{1})$ (cf. [15]), where $H_{0}^{1} (Ω_{1}; Γ_{0}) : = {u \in H_{0}^{1} (Ω_{1}) : u_{| Γ_{0}} = 0}$ , so that the proof of Theorem 2.1 is preserved provided that u has a “good” regularity. To this end, we provide the asymptotic expansion as $x \to O$ of the eigenfunction u (see Sections 8.1–8.2 for the asymptotic and Theorem 8.1 for the justification when $g = const$ and Section 8.3 for the case $g = 0$ ).

There is a vast literature about the asymptotic behaviour of a solution of the Laplace operator with a Neumann boundary condition in bounded domains with cusp-type irregularities (cf. e.g. [16,18,19]). In the paper [20], the authors have discussed the asymptotic behaviour of the eigenfunctions of the Laplace operator along with Neumann boundary conditions in a bounded domain with a cuspidal point. In the paper [5], the regularity in the space of infinitely smooth functions in the case of cuspidal edges has been discussed while in [14] the authors have investigated the regularity of solution of bi-harmonic operator in domains with cusps. We refer to the monographs [10,17] for a detailed study of elliptic boundary problems in domains with other type of singularities.

In our context, we impose two different type of boundary conditions on $Γ_{1}$ and $Γ_{0}$ so that the ansatz of eigenfunction u of problem (1.5)–(1.6) is made of particular functions depending on the geometry of the domain and the boundary conditions (see Section 8.1). Moreover, we show that all eigenfunctions decay exponentially as $x \to O$ when a homogeneous Dirichlet condition is set on the interior boundary $Γ_{0}$ (see Proposition 8.2). We stress out that the performed procedure remains valid if the boundaries of $Ω_{0}$ and $Ω_{1}$ are two curves with different radii of curvature.

The paper is organized as follows. In Section 2 we introduce the weak formulation of the problem (1.1)–(1.4) and we state the main result, i.e. Theorem 2.1. In Section 3 we discuss the formal asymptotic expansions for the eigenpairs in the case $m \in (0, 1 / 2)$ and we provide the proof of Theorem 2.1. In Sections 4–7 we present the asymptotic expansions of eigenpairs for the remaining values of m. We introduce the problems characterizing the leading and the first-order correction terms. We also give very briefly the justification of the anzätze. In Section 8 we derive and justify the asymptotic expansion of the eigenfunctions of the Laplace operator in $Ω_{1} ∖ O$ along with the homogeneous Neumann condition on $Γ_{1}$ and the non-homogeneous Dirichlet boundary condition on $Γ_{0}$ . Moreover we discuss some open questions.

2. Main result

The variational formulation of problem (1.1)–(1.4) reads: find $λ^{ε} \in R$ and ${u_{0}^{ε}, u_{1}^{ε}} \in H^{1} (Ω) ∖ {0}$ satisfying $\begin{matrix} (2.1) & {(\nabla_{x} u_{1}^{ε}, \nabla_{x} φ_{1})}_{Ω_{1}} + ε^{- 1} {(\nabla_{x} u_{0}^{ε}, \nabla_{x} φ_{0})}_{Ω_{0}} = λ^{ε} ({(u_{1}^{ε}, φ_{1})}_{Ω_{1}} + ε^{- 2 m} {(u_{0}^{ε}, φ_{0})}_{Ω_{0}}), \end{matrix}$ $\forall φ \in H^{1} (Ω)$ . Here, ${(\cdot, \cdot)}_{Ω_{i}}$ denotes the natural inner product of Lebesgue space $L^{2} (Ω_{i})$ , $i = 0, 1$ and $m \in R$ . For each $ε > 0$ the bilinear form on the left-hand side of (2.1) is positive, symmetric and closed in $H^{1} (Ω)$ . Due to compactness of the embeddings $H^{1} (Ω_{i}) ↪ L^{2} (Ω_{i})$ , $i = 0, 1$ , the problem (1.1)–(1.4) is associated with a self-adjoint operator whose spectrum consists of the monotone increasing unbounded sequence of eigenvalues (cf., for example, [4, Theorems 10.1.5 and 10.2.2]) $\begin{matrix} (2.2) & 0 = λ_{1}^{ε} < λ_{2}^{ε} ⩽ \dots ⩽ λ_{n}^{ε} ⩽ \dots \to \infty \end{matrix}$ repeated according to their multiplicity. The corresponding eigenfunctions ${u_{0}^{ε}, u_{1}^{ε}}$ are subject to the orthonormalization conditions $\begin{matrix} (2.3) & {(u_{1, i}^{ε}, u_{1, j}^{ε})}_{Ω_{1}} + ε^{- 2 m} {(u_{0, i}^{ε}, u_{0, j}^{ε})}_{Ω_{0}} = δ_{i, j}, i, j \in N, \end{matrix}$ where $δ_{i, j}$ is the Kronecker symbol.

We state the main result of the present paper which provides the asymptotic and its justification for the eigenvalues $λ^{ε}$ for spectral problem (1.1)–(1.4). More specifically, we are only interested on the leading $λ^{0}$ and the first-order correction terms $λ^{'}$ of the asymptotics.

Theorem 2.1.
Let $λ_{n}^{ε}$ be the n-th eigenvalue of the problem ( 1.1 )–( 1.4 ) and let $λ_{n}^{0}$ and $λ_{n}^{'}$ be the leading and first-order correction terms. For $m \in R$ and for any $N \in N$ there exist $ε_{N, m} > 0$ and $C_{N, m} > 0$ such that the estimate $\begin{matrix} (2.4) & | λ_{n}^{ε} - ε^{α} λ_{n}^{0} - ε^{β} λ_{n}^{'} | ⩽ C_{N, m} ε^{γ}, n = 1, \dots, N, \end{matrix}$ holds for some α, β and γ, depending only on m, and $ε \in (0, ε_{N, m})$ .

The feature of Theorem 2.1 is that estimate (2.4) is valid for any value of $m \in R$ , with different exponents α, β and γ. The appearance of different ansätze of $λ^{ε}$ relies on the presence of the factor $ε^{- 2 m}$ involved in the equation (1.2) which determines the range of the real parameter m where the behaviour of $λ^{ε}$ is different.

In the next section, we will give a detailed proof of Theorem 2.1 only when $m \in (0, 1 / 2)$ since this case is more interesting and complicated due to a re-normalization of the eigenfunctions ${u_{0}^{ε}, u_{1}^{ε}}$ and a non-standard choice of approximate eigenfunctions. For the other values of m, we only the state the ansätze of the eigenpairs and all the changes which are needed to prove estimate (2.4).
3. Proof of Theorem 2.1 in the case $0 < m < 1 / 2$

In this section, we provide the proof of Theorem 2.1 in the case $m \in (0, 1 / 2)$ . First, we deduce the formal asymptotics of the eingenpairs $(λ^{ε}, u^{ε})$ .

3.1. Formal asymptotics in the case $0 < m < 1 / 2$

The orthonormalization condition (2.3) suggests to perform the replacements $\begin{array}{l} (3.1) & v_{1}^{ε} (x) = u_{1}^{ε} (x), x \in Ω_{1}, v_{0}^{ε} (x) = ε^{- m} u_{0}^{ε} (x), x \in Ω_{0} . \end{array}$ Hence, ${v_{0}^{ε}, v_{1}^{ε}}$ satisfy the classical orthonormalization condition in $L^{2} (Ω)$ which does not depend anymore on ε. Taking (3.1) into account, equations (1.1)–(1.2) remain unchanged, while the transmission conditions (1.4) turn into $\begin{array}{l} ε^{m} v_{0}^{ε} (x) = v_{1}^{ε} (x), ε^{m - 1} \partial_{ν_{0}} v_{0}^{ε} (x) = \partial_{ν_{1}} v_{1}^{ε} (x), x \in Γ_{0} . \end{array}$ We look for the asymptotic expansion of eigenfunctions ${v_{0}^{ε}, v_{1}^{ε}}$ in the form $\begin{array}{l} (3.2) & v_{0}^{ε} (x) = ε^{m} v_{0}^{0} (x) + ε^{1 - m} v_{0}^{'} (x) + \dots, x \in Ω_{0}, \\ (3.3) & v_{1}^{ε} (x) = v_{1}^{0} (x) + ε^{2 m} v_{1}^{'} (x) + \dots, x \in Ω_{1}, \end{array}$ where the dots denote the lower-order terms which will be estimated. We accept that the eigenvalue $λ^{ε}$ admits the asymptotic ansatz $\begin{matrix} (3.4) & λ^{ε} = λ^{0} + ε^{2 m} λ^{'} + \dots . \end{matrix}$ By inserting expansions (3.2), (3.3), (3.4) in the spectral problem (1.1)–(1.4), we collect coefficients of the alike powers of ε and gather boundary value problems for $v_{0}^{0}$ , $v_{0}^{'}$ and $v_{1}^{0}$ , $v_{1}^{'}$ .

3.1.1. Problem for $v_{0}^{0}$ and $v_{0}^{'}$

The leading term in (3.2) is a solution of the problem $\begin{array}{l} (3.5) & - Δ_{x} v_{0}^{0} (x) = 0, x \in Ω_{0}, \partial_{ν_{0}} v_{0}^{0} (x) = 0, x \in Γ_{0}, \end{array}$ and hence $v_{0}^{0} = c_{0}$ . At this stage, $c_{0}$ is an arbitrary constant in $R$ . The first-order correction term in (3.2) satisfies the boundary value problem $\begin{array}{l} (3.6) & - Δ_{x} v_{0}^{'} (x) = λ^{0} v_{0}^{0} (x), x \in Ω_{0}, \partial_{ν_{0}} v_{0}^{'} (x) & = \partial_{ν_{0}} v_{1}^{0} (x), x \in Γ_{0} . \end{array}$ From the compatibility condition for inhomogeneous Neumann problem, we determine the constant $c_{0}$ : $\begin{matrix} (3.7) & c_{0} = \frac{1}{λ^{0} | Ω_{0} |} \int_{Γ_{0}} \partial_{ν_{0}} v_{1}^{0} d s_{x}, \end{matrix}$ where $| \cdot |$ stands for Lebesgue measure of a set and $λ^{0} \neq 0$ is an eigenvalue of the problem (3.8)–(3.9).

3.1.2. Problem for $v_{1}^{0}$ and $v_{1}^{'}$

The leading terms in (3.3) and (3.4) satisfy the spectral mixed boundary value problem $\begin{array}{l} (3.8) & - Δ_{x} v_{1}^{0} (x) = λ^{0} v_{1}^{0} (x), x \in Ω_{1}, \\ (3.9) & \partial_{ν_{1}} v_{1}^{0} (x) = 0, x \in Γ_{1}, v_{1}^{0} (x) = 0, x \in Γ_{0} . \end{array}$ The variational setting implies the integral identity $\begin{matrix} {(\nabla_{x} v_{1}^{0}, \nabla_{x} φ)}_{Ω_{1}} = λ^{0} {(v_{1}^{0}, φ)}_{Ω_{1}}, φ \in H_{0}^{1} (Ω_{1}, Γ_{0}), \end{matrix}$ where $H_{0}^{1} (Ω_{1}, Γ_{0}) : = {u \in H^{1} (Ω_{1}) : u_{| Γ_{0}} = 0}$ . The spectrum of problem (3.8)–(3.9) is discrete and turns into a monotone unbounded sequences of eigenvalues $\begin{matrix} (3.10) & 0 < λ_{1}^{0} < λ_{2}^{0} ⩽ \dots ⩽ λ_{n}^{0} ⩽ \dots \to + \infty, \end{matrix}$ and the corresponding eigenfunctions $v_{1, 1}^{0}, v_{1, 2}^{0}, \dots$ are subject to the orthonormalization conditions $\begin{matrix} (3.11) & {(v_{1, i}^{0}, v_{1, j}^{0})}_{Ω_{1}} = δ_{i, j}, i, j \in N . \end{matrix}$

The correction term in (3.3) is determined by the boundary value problem $\begin{array}{l} (3.12) & - Δ_{x} v_{1}^{'} (x) - λ^{0} v_{1}^{'} (x) = λ^{'} v_{1}^{0} (x), x \in Ω_{1}, \\ (3.13) & \partial_{ν_{1}} v_{1}^{'} (x) = 0, x \in Γ_{1}, v_{1}^{'} (x) = v_{0}^{0} (x), x \in Γ_{0} . \end{array}$ Since $v_{0}^{0} = c_{0}$ is fixed and defined by (3.7), the boundary condition (3.13) becomes $v_{1}^{'} (x) = c_{0}$ , $x \in Γ_{0}$ . The correction term $λ^{'}$ is determined through the compatibility condition in the problem (3.12)–(3.13). First, we assume that the eigenvalue $λ_{n}^{0} \neq 0$ of problem (3.8)–(3.9) is simple. Then the problem (3.12)–(3.13) has a unique solution if and only if $\begin{array}{l} λ_{n}^{'} \int_{Ω_{1}} {| v_{1, n}^{0} (x) |}^{2} d x = c_{0} \int_{Γ_{0}} \partial_{ν_{0}} v_{0, n}^{'} (x) d s_{x} = - c_{0} \int_{Ω_{0}} Δ_{x} v_{0, n}^{'} (x) d x = c_{0}^{2} λ_{n}^{0} | Ω_{0} | . \end{array}$ Thus, the perturbation term in the ansatz (3.4) takes the form $\begin{matrix} (3.14) & λ_{n}^{'} = c_{0}^{2} λ_{n}^{0} | Ω_{0} | = \frac{1}{λ_{n}^{0} | Ω_{0} |} {(\int_{Γ_{0}} \partial_{ν_{0}} v_{1}^{0} d s_{x})}^{2} . \end{matrix}$

Multiple eigenvalues. In the case $λ_{n}^{0} \neq 0$ is a multiple eigenvalue with multiplicity $τ > 1$ , i.e. $\begin{matrix} (3.15) & λ_{n - 1}^{0} < λ_{n}^{0} = λ_{n + 1}^{0} = \dots = λ_{n + τ - 1}^{0} < λ_{n + τ}^{0}, \end{matrix}$ the expansions (3.2)–(3.3) are still valid. However we predict that the leading terms of $v_{1, n}^{ε}, v_{1, n + 1}^{ε}, \dots$ , $v_{1, n + τ - 1}^{ε}$ are linear combinations of the eigenfunctions $v_{1, n}^{0}, v_{1, n + 1}^{0}, \dots, v_{1, n + τ - 1}^{0}$ of the problem (3.8)–(3.9) associated to eigenvalue $λ_{n}^{0}$ , i.e. $\begin{matrix} (3.16) & V_{1, j}^{0} (x) = a_{n}^{j} v_{1, n}^{0} (x) + \dots + a_{n + τ - 1}^{j} v_{1, n + τ - 1}^{0} (x), j = n, \dots, n + τ - 1 . \end{matrix}$ Furthermore, we require that the columns $\begin{matrix} a^{j} = {(a_{n}^{j}, \dots, a_{n + τ - 1}^{j})}^{⊤} \in R^{τ}, j = n, \dots, n + τ - 1, \end{matrix}$ satisfy the orthonormalization conditions $\begin{matrix} (3.17) & (a^{j}, a^{i}) : = \sum_{k = n}^{n + τ - 1} a_{k}^{j} a_{k}^{i} = δ_{j, i}, j, i = n, \dots, n + τ - 1 . \end{matrix}$ As a consequence, the linear combinations (3.16) with $j = n, \dots, n + τ - 1$ are a new orthonormal basis in the eigenspace of the eigenvalue $λ_{n}^{0}$ .

Bearing in mind the linear combinations (3.16), the compatibility conditions in the problem (3.6) yield the new constant leading terms $v_{0, n}^{0}, \dots, v_{0, n + τ - 1}^{0}$ of the ansatz (3.2) $\begin{matrix} (3.18) & v_{0, j}^{0} = \frac{1}{λ_{n}^{0} | Ω_{0} |} \sum_{k = n}^{n + τ - 1} a_{k}^{j} \int_{Γ_{0}} \partial_{ν_{0}} v_{1, k}^{0} d s_{x}, j = 1, \dots, n + τ - 1 . \end{matrix}$ The correction term $V_{1, j}^{'}$ is determined from the problem $\begin{array}{l} (3.19) & - Δ_{x} V_{1, j}^{'} (x) - λ_{n}^{0} V_{1, j}^{'} (x) = λ_{j}^{'} V_{1, j}^{0} (x), x \in Ω_{1}, \\ (3.20) & \partial_{ν_{1}} V_{1, j}^{'} (x) = 0, x \in Γ_{1}, V_{1, j}^{'} (x) = v_{0, j}^{0}, x \in Γ_{0} . \end{array}$ The Fredholm alternative leading to the necessary and sufficient condition for $V_{1, j}^{'}$ , $j = n, \dots$ , $n + τ - 1$ to exist, is given by $\begin{matrix} λ_{j}^{'} {(V_{1, j}^{0}, v_{1, p}^{0})}_{Ω_{1}} = \int_{Γ_{0}} V_{1, j}^{'} \partial_{ν_{0}} v_{1, p}^{0} (x) d s_{x}, p = n, \dots, n + τ - 1 . \end{matrix}$ Owing to (3.18) and the orthonormalization condition (3.11), the above formulas become $\begin{matrix} (3.21) & λ_{j}^{'} a_{p}^{j} = \sum_{k = n}^{n + τ - 1} a_{k}^{j} \frac{1}{λ_{n}^{0} | Ω_{0} |} \int_{Γ_{0}} \partial_{ν_{0}} v_{1, k}^{0} (x) d s_{x} \int_{Γ_{0}} \partial_{ν_{0}} v_{1, p}^{0} (x) d s_{x}, p = n \dots, n + τ - 1 . \end{matrix}$ We represent the relations (3.21) as an algebraic spectral system $\begin{matrix} M a^{j} = λ_{j}^{'} a^{j}, j = n, \dots, n + τ - 1, \end{matrix}$ with the matrix $M$ of size $τ \times τ$ defined by $\begin{matrix} M_{p k} = \frac{1}{λ_{n}^{0} | Ω_{0} |} \int_{Γ_{0}} \partial_{ν_{0}} v_{1, p}^{0} (x) d s_{x} \int_{Γ_{0}} \partial_{ν_{0}} v_{1, k}^{0} (x) d s_{x}, p, k = n, \dots, n + τ - 1 . \end{matrix}$ It is clear that $M$ is a symmetric matrix, i.e. $M_{p k} = M_{k p}$ . Therefore, it has τ real eigenvalues, $λ_{n}^{'}, λ_{n + 1}^{'}, \dots, λ_{n + τ - 1}^{'}$ , with eigenvectors $a^{n}, a^{n + 1}, \dots, a^{n + τ - 1}$ satisfying the orthonormalization conditions (3.17). It is easy to show that the matrix $M$ has rank 1, since the determinant of the matrix $M$ and all its minors of order k, for $1 < k ⩽ τ - 1$ , are equal to 0. Therefore, the characteristic polynomial of $M$ is simple. Since the determinant of the matrix $M$ and all its minors of order k, for $1 < k ⩽ τ - 1$ , are equal to 0, the characteristic polynomial of $M$ is simply $\begin{matrix} (3.22) & {(λ^{'})}^{τ} - tr (M) {(λ^{'})}^{τ - 1} = 0, \end{matrix}$ with $tr (M)$ being the trace of the matrix $M$ . It follows that the roots of (3.22) $λ_{j}^{'}$ , for $j = n, \dots, n + τ - 1$ , are given by $\begin{matrix} (3.23) & λ_{n}^{'} = \dots = λ_{n + τ - 2}^{'} = 0, λ_{n + τ - 1}^{'} = tr (M) = \frac{1}{λ_{n}^{0} | Ω_{0} |} \sum_{k = n}^{n + τ - 1} {(\int_{Γ_{0}} \partial_{ν_{0}} v_{1, k}^{0} (x) d s_{x})}^{2} . \end{matrix}$

3.1.3. Final remarks

The asymptotic procedure described above can be continued to construct infinite asymptotic series for eigenvalues and eigenfunctions of the problem (1.1)–(1.4). If the eigenvalues $λ_{n}^{0}$ is simple, the analysis just repeats the explained steps and provides the formal series $\begin{matrix} (3.24) & \sum_{j, k = 0}^{\infty} ε^{j m + k (1 - 2 m)} λ_{n}^{(j, k)}, \end{matrix}$ and the difference between the true eigenvalue $λ^{ε}$ and the partial sum of the series (3.24) can be estimated in a way quite similar to Section 3.

The same can be readily done in the case $τ = 2$ when the correction term $λ_{n + 1}^{'}$ in (3.23) does not vanish so that both the eigenvalues $λ_{n}^{ε}$ and $λ_{n + 1}^{ε}$ become simple and therefore can be examined independently. However, if $λ_{n}^{0}$ has multiplicity $τ > 2$ or $τ = 2$ with $λ_{n + 1}^{'} = 0$ (cf. (3.23)), the coefficients of the linear combination (3.16) are not completely determined. In order to compute them, the coefficients $a_{n}^{j}, \dots, a_{n + τ - 1}^{j}$ are assumed to be a linear combination of the eigencolumns associated to the eigenvalue 0 of the matrix $M$ , obtaining the coefficients and the next term of the expansion of $λ^{ε}$ . Nevertheless, there is no argument ensuring that the new matrix has distinct eigenvalues and hence the coefficients of linear combination of $a_{n}^{j}, \dots, a_{n + τ - 1}^{j}$ can not be uniquely defined, so that an iteration of the previous procedure is needed again.

3.2. Justification of asymptotics in the case $m \in (0, 1 / 2)$

3.2.1. Step 1: Convergence theorem

In this subsection, we show that for fixed $n \in N$ the eigenvalue $λ_{n}^{ε}$ converges to $λ_{n}^{0}$ , as $ε \to 0$ , and the corresponding eigenfunctions converge strongly in $L^{2} (Ω_{1})$ .

Proposition 3.1.
The eigenvalues $λ_{n}^{ε}$ of the problem ( 1.1 )–( 1.4 ) and the eigenvalues $λ_{n}^{0}$ of the problem ( 3.8 )–( 3.9 ) are related by passing to the limit $\begin{matrix} λ_{n}^{ε} \to λ_{n}^{0}, as ε \to 0, n \in N . \end{matrix}$

We begin to show the following lemma.
Lemma 3.2.
Assume that for any $n \in N$ there exist $ε_{n} > 0$ and $C_{n} > 0$ such that $\begin{matrix} (3.25) & 0 < λ_{n}^{ε} ⩽ C_{n} for ε \in (0, ε_{n}) . \end{matrix}$ Then, we have that $λ_{n}^{ε} \to λ_{\bar{n}}^{0}$ , for some $\bar{n} \in N$ , as $ε \to 0$ .
Proof.
By virtue of estimate (3.25), whose proof will be given in Remark 3.4, we extract an infinitesimal positive sequence ${ε_{k}}_{k \in N}$ such that $\begin{matrix} (3.26) & λ_{n}^{ε_{k}} \to λ_{\bar{n}}^{0}, ε_{k} \to 0 . \end{matrix}$ In order to simplify the notation we write $λ_{n}^{ε}$ in place of $λ_{n}^{ε_{k}}$ . The normalization condition (2.3), the estimate (3.25) and the weak formulation (2.1) of the spectral problem (1.1)–(1.4) yield $\begin{matrix} {‖ \nabla_{x} u_{1, n}^{ε} ‖}_{Ω_{1}}^{2} + ε^{- 1} {‖ \nabla_{x} u_{0, n}^{ε} ‖}_{Ω_{0}}^{2} = λ_{n}^{ε} ⩽ C_{n} . \end{matrix}$ As a consequence, $\begin{array}{l} {‖ \nabla_{x} u_{1, n}^{ε} ‖}_{Ω_{1}}^{2} & ⩽ C_{n}, {‖ u_{1, n}^{ε} ‖}_{Ω_{1}}^{2} ⩽ 1 . \end{array}$ The norms $‖ u_{1, n}^{ε} ‖_{H^{1} (Ω_{1})}$ are uniformly bounded in $ε \in (0, ε_{n})$ for a fixed n. Then, up to subsequence, $u_{1, n}^{ε}$ converges weakly in $H_{0}^{1} (Ω_{1}, Γ_{0})$ and strongly in $L^{2} (Ω_{1})$ to some function $g_{1}^{0}$ , which can be identified as an eigenfunction $u_{1, \bar{n}}^{0}$ associated to $λ_{\bar{n}}^{0}$ . In fact, if we take an arbitrary function $φ_{1} \in H_{0}^{1} (Ω_{1}, Γ_{0})$ , and $φ_{0} = 0$ in $Ω_{0}$ as a test functions in the integral identity (2.1), it admits the limit passage as $ε \to 0$ , yielding the integral identity $\begin{matrix} (3.27) & {(\nabla_{x} g_{1}^{0}, \nabla_{x} φ_{1})}_{Ω_{1}} = λ_{\bar{n}}^{0} {(g_{1}^{0}, φ_{1})}_{Ω_{1}} . \end{matrix}$ The equality (3.27) gives rise to the problem $\begin{array}{l} - Δ_{x} g_{1}^{0} (x) = λ_{\bar{n}}^{0} g_{1}^{0} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} g_{1}^{0} (x) = 0, x \in Γ_{1}, g_{1}^{0} (x) = 0, x \in Γ_{0}, \end{array}$ which implies that $g_{1}^{0} = u_{1, \bar{n}}^{0}$ . In other terms, $λ_{\bar{n}}^{0}$ is an eigenvalue of the limit problem (3.8)–(3.9) with corresponding eigenfunction $u_{1, \bar{n}}^{0}$ . Concerning the function $u_{0, n}^{ε}$ , we find that $\begin{matrix} ε^{- 1} {‖ \nabla_{x} u_{0, n}^{ε} ‖}_{Ω_{0}}^{2} ⩽ C, ε^{- 2 m} {‖ u_{0, n}^{ε} ‖}_{Ω_{0}}^{2} ⩽ 1, \end{matrix}$ so that $u_{0, n}^{ε}$ converges to 0 strongly in $H_{0}^{1} (Ω_{0})$ and hence in $L^{2} (Ω_{0})$ (if necessary, we can again pass to a subsequence).

The eigenfunction $u_{1, \bar{n}}^{0}$ is also normalized in $L^{2}$ -norm. Indeed, bearing in mind the replacement (3.1), we deduce that $\begin{matrix} {‖ v_{0, n}^{ε} ‖}_{Ω_{0}}^{2} ⩽ 1, {‖ \nabla_{x} v_{0, n}^{ε} ‖}_{Ω_{0}}^{2} ⩽ ε^{1 - 2 m} C, \end{matrix}$ from which it follows that $v_{0, n}^{ε}$ converges strongly in $L^{2} (Ω_{0})$ to some constant $\tilde{c}$ . In order to prove that $\tilde{c} = 0$ , we take $φ_{1} = φ_{0} = ε^{m} \tilde{c}$ as test functions in (2.1), obtaining $\begin{matrix} 0 = λ_{n}^{ε} (ε^{m} \tilde{c} \int_{Ω_{1}} u_{1, n}^{ε} d x + \tilde{c} \int_{Ω_{0}} v_{0, n}^{ε} d x) . \end{matrix}$ Passing to the limit as $ε \to 0$ , we find that $\tilde{c} = 0$ . As a consequence, $ε^{- 2 m} ‖ u_{0, n}^{ε} ‖_{Ω_{0}}^{2} \to 0$ , as $ε \to 0$ and the normalization condition (2.3) leads to $‖ u_{1, \bar{n}}^{0} ‖_{L^{2} (Ω_{1})} = 1$ . □

The goal of the next subsection is to check that $n = \bar{n}$ , concluding hence the proofs of Proposition 3.1 and of Theorem 2.1.
3.2.2. Step 2: Lemma about near eigenvalues and eigenfunctions

Let $H_{ε}$ denote the Hilbert space $H^{1} (Ω)$ endowed with the inner product $\begin{matrix} (3.28) & {⟨ U, V ⟩}_{ε} = {(\nabla_{x} U_{1}, \nabla_{x} V_{1})}_{Ω_{1}} + ε^{- 1} {(\nabla_{x} U_{0}, \nabla_{x} V_{0})}_{Ω_{0}} + {(U_{1}, V_{1})}_{Ω_{1}} + ε^{- 2 m} {(U_{0}, V_{0})}_{Ω_{0}} . \end{matrix}$ We introduce the operator $K_{ε}$ in $H_{ε}$ by the formula $\begin{matrix} (3.29) & {⟨ K_{ε} U, V ⟩}_{ε} = {(U_{1}, V_{1})}_{Ω_{1}} + ε^{- 2 m} {(U_{0}, V_{0})}_{Ω_{0}} \forall U, V \in H_{ε}, \end{matrix}$ and the new spectral parameter $\begin{matrix} (3.30) & k_{n}^{ε} = {(1 + λ_{n}^{ε})}^{- 1} . \end{matrix}$ It is easy to check that $K_{ε}$ is a continuous, self-adjoint, positive and compact operator. Thus, the spectrum of operator $K_{ε}$ consists of the essential spectrum $σ_{ess} (K_{ε}) = {0}$ and an infinitesimal positive sequence of real eigenvalues $\begin{matrix} k_{1}^{ε} ⩾ k_{2}^{ε} ⩾ \dots ⩾ k_{n}^{ε} ⩾ \dots \to 0 . \end{matrix}$ Taking into account formulas (3.28)–(3.30), the integral identity (2.1) is equivalent to the abstract equation $\begin{matrix} K_{ε} U^{ε} = k_{n}^{ε} U^{ε} . \end{matrix}$ The following statement is known as “lemma about near eigenvalues and eigenvectors” (cf. [24]) and follows from the spectral decomposition of the resolvent, cf. [4, Chapter 6].

Lemma 3.3.
Assume $U^{ε} \in H_{ε}$ and $k^{ε} \in R_{+}$ such that $\begin{matrix} {‖ U^{ε} ‖}_{H_{ε}} = 1, {‖ K_{ε} U^{ε} - k^{ε} U^{ε} ‖}_{H_{ε}} = : δ^{ε} \in (0, k^{ε}) . \end{matrix}$ Then in the segment $[k^{ε} - δ^{ε}, k^{ε} + δ^{ε}]$ there is at least one eigenvalue of the operator $K_{ε}$ . Moreover, for any $δ_{ε}^{'} \in (δ^{ε}, k^{ε})$ , there exist coefficients $a_{J^{ε}}^{ε}, \dots, a_{J^{ε} + K^{ε} - 1}^{ε}$ such that $\begin{matrix} {‖ U^{ε} - \sum_{j = J^{ε}}^{J^{ε} + K^{ε} - 1} a_{j}^{ε} u_{j}^{ε} ‖}_{H_{ε}} ⩽ 2 \frac{δ^{ε}}{δ_{ε}^{'}}, \sum_{j = J^{ε}}^{J^{ε} + K^{ε} - 1} {| a_{j}^{ε} |}^{2} = 1, \end{matrix}$ where $u_{J^{ε}}^{ε}, \dots, u_{J^{ε} + K^{ε} - 1}^{ε}$ are eigenvectors associated to all eigenvalues $k_{J^{ε}}^{ε}, \dots, k_{J^{ε} + K^{ε} - 1}^{ε}$ of the operator $K_{ε}$ situated in $[k^{ε} - δ^{ε}, k^{ε} + δ^{ε}]$ . The eigenvectors are subject to the orthonormalization conditions $\begin{matrix} (3.31) & {⟨ u_{i}^{ε}, u_{j}^{ε} ⟩}_{ε} = δ_{i, j} . \end{matrix}$

In the case of a simple eigenvalue $λ_{\bar{n}}^{0}$ of the problem (3.8)–(3.9), the approximate eigenvalue $k_{\bar{n}}^{ε}$ is $\begin{matrix} (3.32) & {(1 + λ_{\bar{n}}^{0} + ε^{2 m} λ_{\bar{n}}^{'})}^{- 1}, \end{matrix}$ where $λ_{\bar{n}}^{'}$ is the asymptotic correction (3.14) and the approximate eigenfunction $U_{\bar{n}}^{ε} = (U_{0, \bar{n}}^{ε}, U_{1, \bar{n}}^{ε})$ is defined by $\begin{matrix} (3.33) & (ε^{2 m} c_{0, \bar{n}} + ε u_{0, \bar{n}}^{'} + ε^{2 - 2 m} U_{0, \bar{n}}^{ε}, u_{1, \bar{n}}^{0} + ε^{2 m} u_{1, \bar{n}}^{'} + ε U_{1, \bar{n}} + ε^{2 - 2 m} U_{1, \bar{n}}^{'}), \end{matrix}$ where $c_{0, \bar{n}}$ is given by (3.7), $u_{0, \bar{n}}^{'}$ is the solution to the problem (3.6), $u_{1, \bar{n}}^{0}$ solves the limit problem (3.8)–(3.9) and $u_{1, \bar{n}}^{'}$ is characterized by the problem (3.12)–(3.13). The arbitrary (but fixed) functions $U_{1, \bar{n}}$ , $U_{1, \bar{n}}^{'}$ in $H^{1} (Ω_{1})$ are such that $\begin{matrix} U_{1, \bar{n}} (x) = u_{0, \bar{n}}^{'} (x), U_{1, \bar{n}}^{'} (x) = U_{0, \bar{n}}^{ε} (x), x \in Γ_{0}, \end{matrix}$ and $U_{0, \bar{n}}^{ε}$ is the solution to the Neumann problem for the Helmholtz operator $\begin{array}{l} (3.34) & - Δ_{x} U_{0, \bar{n}}^{ε} (x) - ε^{1 - 2 m} λ_{\bar{n}}^{0} U_{0, \bar{n}}^{ε} (x) = λ_{\bar{n}}^{0} u_{0, \bar{n}}^{'} (x), x \in Ω_{0}, \\ (3.35) & \partial_{ν_{0}} U_{0, \bar{n}}^{ε} (x) = 0, x \in Γ_{0} . \end{array}$ Denoting by $L_{⊥}^{2} (Ω_{0})$ the subspace ${u \in L^{2} (Ω_{0}) : \int_{Ω_{0}} u (x) d x = 0}$ and setting $H_{⊥}^{2} (Ω_{0}) = H^{2} (Ω_{0}) \cap L_{⊥}^{2} (Ω_{0})$ , the Neumann Laplacian $Δ_{x} : H_{⊥}^{2} (Ω_{0}) \to L_{⊥}^{2} (Ω_{0})$ is an isomorphism. Consequently, for small $ε > 0$ the mapping $- Δ_{x} - ε^{1 - 2 m} λ_{\bar{n}}^{0} Id$ is also an isomorphism, i.e. $U_{0, \bar{n}}^{ε}$ is the unique solution to the problem (3.34)–(3.35) (cf., e.g., [9, Theorem 3.6.1]). Furthermore, the estimate $\begin{matrix} {‖ U_{0, \bar{n}}^{ε} ‖}_{H_{⊥}^{2} (Ω_{0})} ⩽ c λ_{n}^{0} {‖ u_{0, \bar{n}}^{'} ‖}_{L_{⊥}^{2} (Ω_{0})} \end{matrix}$ holds, where the constant c is independent of the parameter ε.

If $λ_{\bar{n}}^{0}$ is a multiple eigenvalue (cf. (3.15)) and $λ_{\bar{n}}^{'}$ in (3.32) is given by (3.23), then the functions $u_{1, j}^{0}$ , $c_{0, j}$ , $u_{1, j}^{'}$ in (3.33) are replaced with $V_{1, j}^{0}$ , $v_{0, j}^{0}$ defined by the formulas (3.16), (3.18) and the solution $V_{1, j}^{'}$ to the problem (3.19)–(3.20) for any $j = \bar{n}, \dots, \bar{n} + τ - 1$ .

The almost eigenfunction (sometimes, also called quasimode) $U_{\bar{n}}^{ε}$ belongs to Hilbert space $H_{ε}$ but in generally it does not satisfy the normalization condition. Then, we apply Lemma 3.3 with $‖ U_{\bar{n}}^{ε} ‖_{H_{ε}}^{- 1} U_{\bar{n}}^{ε} \in H_{ε}$ . Note that for sufficiently small ε the estimate $\begin{matrix} (3.36) & {‖ U_{\bar{n}}^{ε} ‖}_{H_{ε}} ⩾ \frac{1}{2}, \end{matrix}$ follows from formula (3.37). Indeed, the inner product $\begin{array}{l} {⟨ U_{i}^{ε}, U_{j}^{ε} ⟩}_{ε} = & {(\nabla_{x} U_{1, i}^{ε}, \nabla_{x} U_{1, j}^{ε})}_{Ω_{1}} + ε^{- 1} {(\nabla_{x} U_{0, i}^{ε}, \nabla_{x} U_{0, j}^{ε})}_{Ω_{0}} + {(U_{1, i}^{ε}, U_{1, j}^{ε})}_{Ω_{1}} \\ + ε^{- 2 m} {(U_{0, i}^{ε}, U_{0, j}^{ε})}_{Ω_{0}} \\ = & {(\nabla_{x} u_{1, i}^{0}, \nabla_{x} u_{1, j}^{0})}_{Ω_{1}} + {(u_{1, i}^{0}, u_{1, j}^{0})}_{Ω_{1}} + O (ε^{2 m}) \\ = & (1 + λ_{i}^{0}) {(u_{1, i}^{0}, u_{1, j}^{0})}_{Ω_{1}} + O (ε^{2 m}) \\ (3.37) & = & (1 + λ_{i}^{0}) δ_{i j} + O (ε^{2 m}), i, j = 1, 2, \dots \end{array}$ where the last equality is due to orthonormalization conditions (3.11). Note that $O (ε^{2 m})$ contains the terms listed below multiplied by some power of ε and they can be easily estimated: $\begin{array}{l} ε^{2 m} : {(\nabla_{x} u_{1, i}^{0}, \nabla_{x} u_{1, j}^{'})}_{Ω_{1}} + {(\nabla_{x} u_{1, i}^{'}, \nabla_{x} u_{1, j}^{0})}_{Ω_{1}} + {(u_{1, i}^{0}, u_{1, j}^{'})}_{Ω_{1}} + {(u_{1, i}^{'}, u_{1, j}^{0})}_{Ω_{1}} + {(c_{0, i}, c_{0, j})}_{Ω_{0}}; \\ ε^{4 m} : {(\nabla_{x} u_{1, i}^{'}, \nabla_{x} u_{1, j}^{'})}_{Ω_{1}} + {(u_{1, i}^{'}, u_{1, j}^{'})}_{Ω_{1}}; ε^{4 - 6 m} : {(U_{0, i}^{ε}, U_{0, j}^{ε})}_{Ω_{0}}; \\ ε : {(\nabla_{x} u_{1, i}^{0}, \nabla_{x} U_{1, j})}_{Ω_{1}} + {(\nabla_{x} U_{1, i}, \nabla_{x} u_{1, j}^{0})}_{Ω_{1}} + {(\nabla_{x} u_{0, i}^{'}, \nabla_{x} u_{0, j}^{'})}_{Ω_{0}} \\ + {(u_{1, i}^{0}, U_{1, j})}_{Ω_{1}} + {(U_{1, i}, u_{1, j}^{0})}_{Ω_{1}}; \\ ε^{2 m + 1} : {(\nabla_{x} u_{1, i}^{'}, \nabla_{x} U_{1, j})}_{Ω_{1}} + {(\nabla_{x} U_{1, i}, \nabla_{x} u_{1, j}^{'})}_{Ω_{1}} + {(u_{1, i}^{'}, U_{1, j})}_{Ω_{1}} + {(U_{1, i}, u_{1, j}^{'})}_{Ω_{1}}; \\ ε^{2 - 2 m} : {(\nabla_{x} u_{1, i}^{0}, \nabla_{x} U_{1, j}^{'})}_{Ω_{1}} + {(\nabla_{x} U_{1, i}^{'}, \nabla_{x} u_{1, j}^{0})}_{Ω_{1}} + {(\nabla_{x} u_{0, i}^{'}, \nabla_{x} U_{0, j}^{ε})}_{Ω_{0}} \\ + {(\nabla_{x} U_{0, i}^{ε}, \nabla_{x} u_{0, j}^{'})}_{Ω_{0}} + {(u_{1, i}^{0}, U_{1, j}^{'})}_{Ω_{1}} + {(U_{1, i}^{'}, u_{1, j}^{0})}_{Ω_{1}} + {(u_{0, i}^{'}, u_{0, j}^{'})}_{Ω_{0}}; \\ ε^{2} : {(\nabla_{x} u_{1, i}^{'}, \nabla_{x} U_{1, j}^{'})}_{Ω_{1}} + {(\nabla_{x} U_{1, i}, \nabla_{x} U_{1, j})}_{Ω_{1}} + {(\nabla_{x} U_{1, i}^{'}, \nabla_{x} u_{1, j}^{'})}_{Ω_{1}} \\ + {(u_{1, i}^{'}, U_{1, j}^{'})}_{Ω_{1}} + {(U_{1, i}, U_{1, j})}_{Ω_{1}} + {(U_{1, i}^{'}, u_{1, j}^{'})}_{Ω_{1}}; \\ ε^{3 - 4 m} : {(\nabla_{x} U_{0, i}^{ε}, \nabla_{x} U_{0, j}^{ε})}_{Ω_{0}} + {(u_{0, i}^{'}, U_{0, j}^{ε})}_{Ω_{0}} + {(U_{0, i}^{ε}, u_{0, j}^{'})}_{Ω_{0}}; \\ ε^{3 - 2 m} : {(\nabla_{x} U_{1, i}, \nabla_{x} U_{1, j}^{'})}_{Ω_{1}} + {(\nabla_{x} U_{1, i}^{'}, \nabla_{x} U_{1, j})}_{Ω_{1}} + {(U_{1, i}, U_{1, j}^{'})}_{Ω_{1}} + {(U_{1, i}^{'}, U_{1, j})}_{Ω_{1}}; \\ ε^{4 - 4 m} : {(\nabla_{x} U_{1, i}^{'}, \nabla_{x} U_{1, j}^{'})}_{Ω_{1}} + {(U_{1, i}^{'}, U_{1, j}^{'})}_{Ω_{1}} . \end{array}$ Consequently, we obtain $\begin{array}{l} δ_{\bar{n}}^{ε} & = {‖ U_{\bar{n}}^{ε} ‖}_{H_{ε}}^{- 1} {‖ K_{ε} U_{\bar{n}}^{ε} - k_{\bar{n}}^{ε} U_{\bar{n}}^{ε} ‖}_{H_{ε}} \\ = {‖ U_{\bar{n}}^{ε} ‖}_{H_{ε}}^{- 1} sup_{\begin{array}{c} W_{ε} \in H_{ε} \\ ‖ W_{ε} ‖_{H_{ε}} = 1 \end{array}} | {⟨ K_{ε} U_{\bar{n}}^{ε} - k_{\bar{n}}^{ε} U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} | \\ = {‖ U_{\bar{n}}^{ε} ‖}_{H_{ε}}^{- 1} (k_{\bar{n}}^{ε}) sup_{\begin{array}{c} W_{ε} \in H_{ε} \\ ‖ W_{ε} ‖_{H_{ε}} = 1 \end{array}} | {(k_{\bar{n}}^{ε})}^{- 1} {⟨ K_{ε} U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} - {⟨ U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} | \\ ⩽ c sup_{\begin{array}{c} W_{ε} \in H_{ε} \\ ‖ W_{ε} ‖_{H_{ε}} = 1 \end{array}} | {(k_{\bar{n}}^{ε})}^{- 1} {⟨ K_{ε} U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} - {⟨ U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} |, \end{array}$ where in the last inequality we used (3.36) and ${(k_{\bar{n}}^{ε})}^{- 1} ⩾ 1$ . Now, we focus only on the absolute value. Using formulas (3.28) and (3.29), we find $\begin{array}{l} | {(k_{\bar{n}}^{ε})}^{- 1} {⟨ K_{ε} U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} - {⟨ U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} | = & | J_{0} + ε^{2 m} J_{1} + ε^{1 - 2 m} J_{2} + ε^{4 m} λ_{\bar{n}}^{'} {(u_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Ω_{1}} \\ + ε^{2 - 4 m} λ_{\bar{n}}^{0} {(U_{0, \bar{n}}^{ε}, W_{0}^{ε})}_{Ω_{0}} + ε J_{3} + ε^{1 + 2 m} λ_{\bar{n}}^{'} {(U_{1, \bar{n}}, W_{1}^{ε})}_{Ω_{1}} \\ (3.38) & \times ε^{2 - 2 m} J_{4} + ε^{2} λ_{\bar{n}}^{'} {(U_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Ω_{1}} |, \end{array}$ where we have set $\begin{array}{l} J_{0} : = λ_{\bar{n}}^{0} {(u_{1, \bar{n}}^{0}, W_{1}^{ε})}_{Ω_{1}} + λ_{\bar{n}}^{0} {(c_{0, \bar{n}}, W_{0}^{ε})}_{Ω_{0}} - {(\nabla_{x} u_{1, \bar{n}}^{0}, \nabla_{x} W_{1}^{ε})}_{Ω_{1}} - {(\nabla_{x} u_{0, \bar{n}}^{'}, \nabla_{x} W_{0}^{ε})}_{Ω_{0}}; \\ J_{1} : = λ_{\bar{n}}^{'} {(c_{0, \bar{n}}, W_{0}^{ε})}_{Ω_{0}} + λ_{\bar{n}}^{0} {(u_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Ω_{1}} + λ_{\bar{n}}^{'} {(u_{1, \bar{n}}^{0}, W_{1}^{ε})}_{Ω_{1}} - {(\nabla_{x} u_{1, \bar{n}}^{'}, \nabla_{x} W_{1}^{ε})}_{Ω_{1}}; \\ J_{2} : = λ_{\bar{n}}^{0} {(u_{0, \bar{n}}^{'}, W_{0}^{ε})}_{Ω_{0}} - {(\nabla_{x} U_{0, \bar{n}}^{ε}, \nabla_{x} W_{0}^{ε})}_{Ω_{0}}; \\ J_{3} : = λ_{\bar{n}}^{0} {(U_{1, \bar{n}}, W_{1}^{ε})}_{Ω_{1}} + λ_{\bar{n}}^{'} {(u_{0, \bar{n}}^{'}, W_{0}^{ε})}_{Ω_{0}} - {(\nabla_{x} U_{1, \bar{n}}, \nabla_{x} W_{1}^{ε})}_{Ω_{1}}; \\ J_{4} : = λ_{\bar{n}}^{0} {(U_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Ω_{1}} + λ_{\bar{n}}^{'} {(U_{0, \bar{n}}^{ε}, W_{0}^{ε})}_{Ω_{0}} - {(\nabla_{x} U_{1, \bar{n}}^{'}, \nabla_{x} W_{1}^{ε})}_{Ω_{1}} . \end{array}$ Integrating by parts the problems (3.8)–(3.9), (3.6) and (3.12)–(3.13), we obtain that the term $J_{0}$ vanishes and $J_{1}$ turns into ${\tilde{J}}_{1}$ given by $\begin{matrix} {\tilde{J}}_{1} : = λ_{\bar{n}}^{'} {(c_{0, \bar{n}}, W_{0}^{ε})}_{Ω_{0}} - {(\partial_{ν_{0}} u_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Γ_{0}} . \end{matrix}$

Therefore, (3.38) becomes $\begin{array}{l} | {(k_{\bar{n}}^{ε})}^{- 1} {⟨ K_{ε} U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} - {⟨ U_{\bar{n}}^{ε}, W^{ε} ⟩}_{ε} | = & | ε^{2 m} {\tilde{J}}_{1} + ε^{1 - 2 m} J_{2} + ε^{4 m} λ_{\bar{n}}^{'} {(u_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Ω_{1}} \\ + ε^{2 - 4 m} λ_{\bar{n}}^{0} {(U_{0, \bar{n}}^{ε}, W_{0}^{ε})}_{Ω_{0}} + ε J_{3} + ε^{1 + 2 m} λ_{\bar{n}}^{'} {(U_{1, \bar{n}}, W_{1}^{ε})}_{Ω_{1}} \\ + ε^{2 - 2 m} J_{4} + ε^{2} λ_{\bar{n}}^{'} {(U_{1, \bar{n}}^{'}, W_{1}^{ε})}_{Ω_{1}} | . \end{array}$

Note that $ε^{1 - 2 m} J_{2} + ε^{2 - 4 m} λ_{\bar{n}}^{0} {(U_{0, \bar{n}}^{ε}, W_{0}^{ε})}_{Ω_{0}} = 0$ due to the fact that $U_{0, \bar{n}}^{ε}$ can be written as Neumann series (cf. [9, Theorem 3.6.1]). Moreover, the definition of the inner product (3.28) in the Hilbert space $H_{ε}$ yields the following estimates of the classical norm in $L^{2} (Ω_{i})$ , $i = 0, 1$ $\begin{matrix} {‖ W_{1}^{ε} ‖}_{Ω_{1}} ⩽ ‖ W ‖_{H_{ε}}, {‖ \nabla_{x} W_{1}^{ε} ‖}_{Ω_{1}} ⩽ {‖ W^{ε} ‖}_{H_{ε}}, {‖ W^{ε} ‖}_{Ω_{0}} ⩽ ε^{m} {‖ W^{ε} ‖}_{H_{ε}} . \end{matrix}$ Finally, $\begin{array}{l} δ_{\bar{n}}^{ε} & ⩽ C_{1} ε^{3 m} + C_{2} ε + C_{3} ε^{4 m} + C_{4} ε^{1 + m} + C_{5} ε^{1 + 2 m} + C_{6} ε^{2 - 2 m} + C_{7} ε^{2 - m} + C_{8} ε^{2} ⩽ C ε^{γ}, \end{array}$ where $γ = min {3 m, 1}$ . Then, the first part of Lemma 3.3 implies that there exists at least one eigenvalue $k_{n}^{ε}$ of $K_{ε}$ such that $\begin{matrix} (3.39) & | k_{\bar{n}}^{ε} - k_{n}^{ε} | ⩽ C ε^{γ} . \end{matrix}$ From the definition of $k_{\bar{n}}^{ε}$ and $k_{n}^{ε}$ , it follows that if $C ε^{γ} < \frac{1}{2} {(1 + λ_{\bar{n}}^{0} + λ_{\bar{n}}^{'})}^{- 1}$ , then $λ^{ε} ⩽ 2 (λ_{\bar{n}}^{0} + λ_{\bar{n}}^{'}) + 1$ . Hence, the inequality (3.39) can be written as $\begin{matrix} (3.40) & | λ_{n}^{ε} - λ_{\bar{n}}^{0} - ε^{2 m} λ_{\bar{n}}^{'} | ⩽ C | 1 + λ_{n}^{ε} | | 1 + λ_{\bar{n}}^{0} + ε^{2 m} λ_{\bar{n}}^{'} | ε^{γ} ⩽ C_{\bar{n}} ε^{γ} \forall ε \in (0, ε_{N}) . \end{matrix}$ In order to show (2.4) and to conclude the proof of Theorem 2.1, we must check that the indices n and $\bar{n}$ in the inequality (3.40) coincide. To this end, we will apply the second part of Lemma 3.3.

Assume that $λ_{\bar{n}}^{0}$ is an eigenvalue of multiplicity $τ ⩾ 2$ of the problem (3.8)–(3.9). In view of definition (3.33) of almost eigenfunction associated to $λ_{\bar{n}}^{0}$ , we consider the τ copies of almost eigenfunction $U_{p}^{ε}$ , for $p = \bar{n}, \dots, \bar{n} + τ - 1$ , and we introduce $\begin{matrix} δ_{\bar{n} ε}^{'} = T max {δ_{\bar{n}}^{ε}, \dots, δ_{\bar{n} + τ - 1}^{ε}}, \end{matrix}$ where T is large and fixed (independent of ε). The second part of Lemma 3.3 gives the normalized columns $a_{J^{ε}}^{p ε}, \dots, a_{J^{ε} + K^{ε} - 1}^{p ε}$ satisfying the inequality $\begin{matrix} (3.41) & {‖ U_{p}^{ε} - \sum_{j = J^{ε}}^{J^{ε} + K^{ε} - 1} a_{j}^{p ε} u_{j}^{ε} ‖}_{H_{ε}} ⩽ \frac{2}{T}, p = \bar{n}, \dots, \bar{n} + τ - 1 . \end{matrix}$ We aim to show that in the closure of a $δ_{\bar{n} ε}^{'}$ -neighbourhood of the point $k_{\bar{n}}^{ε}$ there are at least τ eigenvalues of the operator $K_{ε}$ , i.e. in the inequality (3.41) the number $K^{ε}$ is such that $K^{ε} ⩾ τ$ . The equality (3.37) implies the estimate $\begin{matrix} (3.42) & | {⟨ U_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} - (1 + λ_{p}^{0}) δ_{p, q} | ⩽ C ε^{2 m}, p, q = \bar{n}, \dots, \bar{n} + τ - 1 . \end{matrix}$ Set $\begin{matrix} S_{p}^{ε} = \sum_{j = N^{ε}}^{N^{ε} + K^{ε} - 1} a_{j}^{p ε} u_{j}^{ε}, p = \bar{n}, \dots, \bar{n} + τ - 1 . \end{matrix}$ In view of the estimate (3.42) and orthonormalization condition (3.31) of $u^{ε}$ , we find $\begin{array}{l} | \sum_{j = N^{ε}}^{N^{ε} + K^{ε} - 1} a_{j}^{p ε} \overline{a_{j}^{q ε}} - (1 + λ_{\bar{n}}^{0}) δ_{p, q} | \\ = | {⟨ \sum_{j = N^{ε}}^{N^{ε} + K^{ε} - 1} a_{j}^{p ε} u_{j}^{ε}, \sum_{j = N^{ε}}^{N^{ε} + K^{ε} - 1} a_{j}^{q ε} u_{j}^{ε} ⟩}_{ε} - (1 + λ_{\bar{n}}^{0}) δ_{p, q} | \\ = | {⟨ S_{p}^{ε}, S_{q}^{ε} ⟩}_{ε} - {⟨ S_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} + {⟨ S_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} - {⟨ U_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} \\ + {⟨ U_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} - (1 + λ_{\bar{n}}^{0}) δ_{p, q} | \\ ⩽ | {⟨ S_{p}^{ε}, S_{q}^{ε} - U_{q}^{ε} ⟩}_{ε} + {⟨ S_{p}^{ε} - U_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} | + | {⟨ U_{p}^{ε}, U_{q}^{ε} ⟩}_{ε} - (1 + λ_{\bar{n}}^{0}) δ_{p, q} | \\ ⩽ {‖ S_{p}^{ε} ‖}_{ε} {‖ S_{q}^{ε} - U_{q}^{ε} ‖}_{ε} + {‖ S_{p}^{ε} - U_{p}^{ε} ‖}_{ε} {‖ U_{q}^{ε} ‖}_{ε} + O (ε^{2 m}) \\ ⩽ C (ε^{2 m} + \frac{4}{T}) . \end{array}$ We conclude that for sufficiently large T, the columns $a^{p ε}$ turn out to be almost orthonormalized that is possible only if $K^{ε} ⩾ τ$ . Consequently, for τ-multiple eigenvalue $λ_{\bar{n}}^{0}$ there are at least τ distinct eigenvalues $k_{j}^{ε}, \dots, k_{j + τ - 1}^{ε}$ of the operator $H_{ε}$ such that $\begin{matrix} (3.43) & | k_{\bar{n}}^{ε} - k_{j}^{ε} | ⩽ T c_{n} ε^{γ}, j = n, \dots, n + τ - 1 . \end{matrix}$
Remark 3.4.
The formula (3.43) leads to check the inequality (3.25). Indeed, for each eigenvalue $λ_{n}^{0}$ of the sequence (3.10), one can associate the eigenvalue $λ_{M (n)}^{ε}$ such that $λ_{M (n)}^{ε} ⩽ λ_{n}^{0} + C_{n} ε^{γ}$ . Moreover $M (n_{1}) < M (n_{2})$ if $n_{1} < n_{2}$ . Consequently $n ⩽ M (n)$ and $\begin{matrix} λ_{n}^{ε} ⩽ λ_{M (n)}^{ε} ⩽ λ_{n}^{0} + C_{n} ε^{2 m} ⩽ λ_{n}^{0} + C_{n}, \end{matrix}$ which implies (3.25).

In order to conclude the proof of Theorem 2.1, we will check that the eigenvalues $λ_{n}^{ε}, \dots, λ_{n + τ - 1}^{ε}$ of the sequence (2.2) satisfy the estimate (2.4). In other words, the equality $\bar{n} = n$ holds true in (3.40).

Let $\bar{n}$ be some index such that $λ_{\bar{n}}^{0}$ is τ-multiple eigenvalue of the problem (3.8)–(3.9). If we assume $M (\bar{n} + τ - 1) > \bar{n} + τ - 1$ , then there exists an eigenvalue $λ_{J^{ε}}^{ε}$ with $J^{ε} ⩽ M (\bar{n} + τ - 1)$ , such that $\begin{matrix} λ_{J^{ε}}^{ε} ⩽ λ_{\bar{n} + τ - 1}^{0} + ε^{2 m} λ_{\bar{n} + τ - 1}^{'} + C ε^{γ} < λ_{\bar{n} + τ}^{0} . \end{matrix}$ From Lemma 3.2, the eigenpair $(λ_{J^{ε}}^{ε}, U_{J^{ε}}^{ε})$ converges to eigenelement $(λ_{J^{0}}^{}, U^{})$ of the limit problem (3.8)–(3.9), where $U^{}$ is orthogonal to $U_{1}^{0}, \dots, U_{\bar{n} + τ - 1}^{0}$ in $L^{2} (Ω_{1})$ . This last claim is invalid, because of the min-max principle (see, e.g. [4]) $\begin{matrix} λ_{J^{0}}^{} = max_{\begin{array}{c} E \subset H_{0}^{1} (Ω_{1}) \\ dim E = J^{ε} - 1 \end{array}} min_{\begin{array}{c} v \in E^{⊥} \\ v \neq 0 \end{array}} \frac{‖ \nabla_{x} v ‖_{L^{2} (Ω_{1})}}{‖ v ‖_{L^{2} (Ω_{1})}} \end{matrix}$ and the inequality $λ_{J^{0}}^{*} < λ_{\bar{n} + τ}^{0}$ . Thus, $\bar{n} = n$ and Theorem 2.1 is proved.
4. Asymptotic expansion for $m < 0$

We briefly describe the behaviour of the eigenpairs of the problem

We accept that the asymptotic expansion for an eigenvalue $λ^{ε}$ and the corresponding eigenfunction ${u_{0}^{ε}, u_{1}^{ε}}$ takes the form $\begin{array}{l} (4.1) & λ^{ε} = λ^{0} + ε^{min {1, - 2 m}} λ^{'} + \dots, \\ (4.2) & u_{0}^{ε} (x) = u_{0}^{0} (x) + ε u_{0}^{'} (x) + \dots, x \in Ω_{0}, \\ (4.3) & u_{1}^{ε} (x) = u_{1}^{0} (x) + ε^{min {1, - 2 m}} u_{1}^{'} (x) + \dots, x \in Ω_{1}, \end{array}$ where $γ = min {1 - m, 2}$ . Here, the dots denote the lower-order terms. We replace the expansions (4.1)–(4.2) in the problem (1.1)–(1.4) and we collect the coefficients of the same powers of ε. Therefore, we obtain that the leading term in (4.2) is a solution to homogeneous Neumann problem (3.5) with $u_{0}^{0}$ in place of $v_{0}^{0}$ . In other words, $u_{0}^{0}$ is a constant $c_{0}$ . The correction term $u_{0}^{'}$ , defined up to an additive constant, satisfies $\begin{array}{l} - Δ_{x} u_{0}^{'} (x) = 0, x \in Ω_{0}, \partial_{ν_{0}} u_{0}^{'} (x) = \partial_{ν_{0}} u_{1}^{0} (x), x \in Γ_{0} . \end{array}$ The compatibility condition reads $\begin{matrix} (4.4) & \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{0} (x) d s_{x} = 0 . \end{matrix}$ We may assume that $u_{0}^{'} (x)$ is given by ${\tilde{u}}_{0}^{'} + {\hat{u}}_{0}^{'} (x)$ where ${\tilde{u}}_{0}^{'}$ is a constant and ${\hat{u}}_{0}^{'} (x)$ is a function in $H^{1} (Ω_{0})$ such that its integral mean on $Ω_{0}$ is zero.

The leading terms $λ_{n}^{0}$ and $u_{1}^{0}$ in the ansätze (4.1) and (4.3) are obtained from the spectral problem $\begin{array}{l} (4.5) & - Δ_{x} u_{1}^{0} (x) = λ^{0} u_{1}^{0} (x), x \in Ω_{1}, \\ (4.6) & \partial_{ν_{1}} u_{1}^{0} (x) = 0, x \in Γ_{1}, u_{1}^{0} (x) = u_{0}^{0} (x), x \in Γ_{0}, \end{array}$ along with the integral condition (4.4). To write down the variational formulation of the problem (4.5)–(4.6), we set $H_{∙}^{1} (Ω_{1}, Γ_{0})$ as the subspace of functions in $H^{1} (Ω_{1})$ with a constant trace on the boundary $Γ_{0}$ . For $φ \in H_{∙}^{1} (Ω_{1}, Γ_{0})$ , the Green formula provides $\begin{array}{l} - \int_{Ω_{1}} Δ_{x} u_{1}^{0} (x) φ (x) d x & = \int_{Ω_{1}} \nabla_{x} u_{1}^{0} (x) \nabla_{x} φ (x) d x - \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{0} (x) φ (x) d s_{x} \\ = λ^{0} \int_{Ω_{1}} u_{1}^{0} (x) φ (x) d x . \end{array}$ Since φ is a constant on the boundary $Γ_{0}$ , it follows that $\begin{matrix} \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{0} (x) φ (x) d s_{x} = const \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{0} (x) d s_{x} = 0 . \end{matrix}$ As a consequence, the variational formulation of (4.5)–(4.6) reads: find the eigenvalue $λ^{0} \in R$ and the corresponding eigenfunction $u_{1}^{0} \in H_{∙}^{1} (Ω_{1}, Γ_{0}) ∖ {0}$ such that $\begin{matrix} (4.7) & {(\nabla_{x} u_{1}^{0}, \nabla_{x} φ)}_{Ω_{1}} = λ^{0} {(u_{1}^{0}, φ)}_{Ω_{1}} \forall φ \in H_{∙}^{1} (Ω_{1}, Γ_{0}) . \end{matrix}$ Note that the integral equality (4.7) implies the condition (4.4). The problem (4.7) admits the eigenvalues sequence (3.10) and the corresponding eigenfunctions $u_{1}^{0}$ are orthonormalized in $L^{2} (Ω_{1})$ .

In order to compute the correction term $u_{1}^{'}$ , we distinguish three cases depending on m: $m \in (- 1 / 2, 0)$ , $m = - 1 / 2$ and $m \in (- \infty, - 1 / 2)$ . Let us investigate the first case, i.e. $m \in (- 1 / 2, 0)$ . The correction term $u_{1}^{'}$ satisfies the problem $\begin{array}{l} (4.8) & - Δ_{x} u_{1}^{'} (x) - λ^{0} u_{1}^{'} (x) = λ^{'} u_{1}^{0} (x), x \in Ω_{1}, \\ (4.9) & \partial_{ν_{1}} u_{1}^{'} (x) = 0, x \in Γ_{1}, u_{1}^{'} (x) = 0, x \in Γ_{0} . \end{array}$ In order to compute explicitly $λ^{'}$ , we have to involve lower-order terms in the asymptotic (4.2). The term $u_{0}^{″}$ has order $ε^{1 - 2 m}$ and it is a solution to problem $\begin{array}{l} (4.10) & - Δ_{x} u_{0}^{″} (x) = λ^{0} u_{0}^{0}, x \in Ω_{0}, \partial_{ν_{0}} u_{0}^{″} (x) = \partial_{ν_{0}} u_{1}^{'} (x), x \in Γ_{0} . \end{array}$ The compatibility condition is given by $\begin{matrix} (4.11) & \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{'} (x) d s_{x} = λ^{0} c_{0} | Ω_{0} | . \end{matrix}$ If $λ_{n}^{0}$ is a simple eigenvalue, the correction term $λ^{'}$ is given by $\begin{matrix} (4.12) & λ^{'} = - {(\partial_{ν_{0}} u_{1}^{'}, u_{1, n}^{0})}_{Γ_{0}} = - λ^{0} c_{0}^{2} | Ω_{0} | . \end{matrix}$ Suppose now that $λ_{n}^{0}$ is a τ-multiple eigenvalue. As in Section 2, the leading terms of the asymptotics of the eigenfunctions $u_{1, n}^{ε}, \dots, u_{1, n + τ - 1}^{ε}$ are predicted in the form of linear combinations $\begin{matrix} (4.13) & U_{1, j}^{0} (x) = b_{n}^{j} u_{1, n}^{0} (x) + \dots + b_{n + τ - 1}^{j} u_{1, n + τ - 1}^{0} (x), j = n, \dots, n + τ - 1, \end{matrix}$ of the eigenfunctions $u_{1, n}^{0}, \dots, u_{1, n + τ - 1}^{0}$ of the limit problem (4.5)–(4.6). The coefficients $b_{n}^{j}, \dots, b_{n + τ - 1}^{j}$ satisfy the orthonormalization condition (3.17). Therefore, first order corrector $U_{1, j}^{'}$ , for $j = n, \dots, n + τ - 1$ , in the asymptotics (5.3) satisfies the problem $\begin{array}{l} - Δ_{x} U_{1, j}^{'} (x) - λ_{n}^{0} U_{1, j}^{'} (x) = λ_{j}^{'} U_{1, j}^{0} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} U_{1, j}^{'} (x) = 0, x \in Γ_{1}, U_{1, j}^{'} (x) = 0, x \in Γ_{0} . \end{array}$ Following the same arguments of Section 3.1.2, we deduce that $\begin{matrix} (4.14) & λ_{j}^{'} b_{p}^{j} = \sum_{k = n}^{n + τ - 1} b_{k}^{j} \frac{1}{λ_{n}^{0} | Ω_{0} |} \int_{Γ_{0}} \partial_{ν_{0}} u_{1, k}^{'} (x) d s_{x} \int_{Γ_{0}} \partial_{ν_{0}} u_{1, p}^{'} (x) d s_{x}, for p = n, \dots, n + τ - 1 . \end{matrix}$

Let us consider the case $m \in (- \infty, - 1 / 2)$ . In this case, the correction term $u_{1}^{'}$ satisfies $\begin{array}{l} (4.15) & - Δ_{x} u_{1}^{'} (x) - λ^{0} u_{1}^{'} (x) = λ^{'} u_{1}^{0} (x), x \in Ω_{1}, \\ (4.16) & \partial_{ν_{1}} u_{1}^{'} (x) = 0, x \in Γ_{1}, u_{1}^{'} (x) = {\hat{u}}_{0}^{'} (x) + {\tilde{u}}_{0}^{'}, x \in Γ_{0} . \end{array}$ The term $u_{0}^{″}$ is of order $ε^{2}$ and it solves the problem $\begin{array}{l} - Δ_{x} u_{0}^{″} (x) = 0, x \in Ω_{0}, \partial_{ν_{0}} u_{0}^{″} (x) = \partial_{ν_{0}} u_{1}^{'} (x), x \in Γ_{0} . \end{array}$ The compatibility condition reads as $\begin{matrix} (4.17) & \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{'} (x) d s_{x} = 0 . \end{matrix}$ In case of simple eigenvalue $λ_{n}^{0}$ , the correction term $λ_{n}^{'}$ is given by $\begin{array}{l} (4.18) & λ_{n}^{'} = {(\partial_{ν_{0}} u_{1, n}^{0}, u_{1, n}^{'})}_{Γ_{0}} = {(\partial_{ν_{0}} u_{0, n}^{'}, u_{0, n}^{'})}_{Γ_{0}} = - {(\nabla_{x} u_{0, n}^{'}, \nabla_{x} u_{0, n}^{'})}_{Ω_{0}} = - {‖ \nabla_{x} u_{0, n}^{'} ‖}_{L^{2} (Ω_{0})}^{2} . \end{array}$ Suppose, now, that $λ_{n}^{0}$ is a τ-multiple eigenvalue. We predict that the leading terms of the asymptotics of the eigenfunctions $u_{1, n}^{ε}, \dots, u_{1, n + τ - 1}^{ε}$ take the form (4.13). Then, the first order corrector $U_{1, j}^{'}$ in (4.3) satisfies the problem $\begin{array}{l} - Δ_{x} U_{1, j}^{'} (x) - λ_{n}^{0} U_{1, j}^{'} (x) = λ_{j}^{'} U_{1, j}^{0} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} U_{1, j}^{'} (x) = 0, x \in Γ_{1}, U_{1, j}^{'} (x) = \sum_{k = n}^{n + τ - 1} b_{k}^{j} ({\hat{u}}_{0, k}^{'} (x) + {\tilde{u}}_{0, k}^{'}), x \in Γ_{0} . \end{array}$ From the Fredholm alternative and the formula (4.17), we get the τ compatibility conditions $\begin{array}{l} λ_{j}^{'} {(U_{1, j}^{0}, u_{1, q}^{0})}_{Ω_{1}} = {(\partial_{ν_{0}} u_{1, q}^{0}, U_{1, j}^{'})}_{Γ_{0}} = \sum_{k = n}^{n + τ - 1} b_{k}^{j} {(\partial_{ν_{0}} u_{0, q}^{'}, u_{0, k}^{'})}_{Γ_{0}} = - \sum_{k = n}^{n + τ - 1} b_{k}^{j} {(\nabla_{x} u_{0, k}^{'}, \nabla_{x} u_{0, q}^{'})}_{Ω_{0}}, \end{array}$ which imply that $\begin{array}{l} λ_{j}^{'} b_{q}^{j} = - \sum_{k = n}^{n + τ - 1} b_{k}^{j} {(\nabla_{x} u_{0, j}^{'}, \nabla_{x} u_{0, q}^{'})}_{Ω_{0}} . \end{array}$ Such an equality can be written in the form of the linear system of τ algebraic equations $\begin{matrix} (4.19) & G b^{j} = λ_{j}^{'} b^{j}, j = n, \dots, n + τ - 1 . \end{matrix}$ Here, G is the Gram matrix whose entries are $\begin{matrix} G_{i, j} = - {(\nabla_{x} u_{0, i}^{'}, \nabla_{x} u_{0, j}^{'})}_{Ω_{0}}, i, j = n, \dots, n + τ - 1 . \end{matrix}$ Since G is a symmetric $τ \times τ$ matrix, its eigenvalues $λ_{n}^{'}, \dots, λ_{n + τ - 1}^{'}$ are real and negative. Indeed, the derivatives $\partial_{ν_{0}} u_{1, n}^{0}, \dots, \partial_{ν_{0}} u_{1, n + τ - 1}^{0}$ are linearly independent in $L^{2} (Γ_{0})$ . Otherwise, a linear combination $\begin{matrix} U (x) = \sum_{k = n}^{n + τ - 1} a_{i} u_{1, i}^{0} (x), x \in Ω_{1}, \end{matrix}$ satisfies the equation $- Δ_{x} U (x) = λ^{0} U (x)$ , $x \in Ω_{1}$ , and simultaneously two boundary conditions $U (x) = const$ and $\partial_{ν_{0}} U (x) = 0$ , $x \in Γ_{0}$ . This is a contradiction due to the theorem on strong unique continuation (e.g. cf. [25]). Hence, $\nabla_{x} u_{0, n}^{'}, \dots, \nabla_{x} u_{0, n + τ - 1}^{'}$ are linearly independent in $L^{2} {(Ω_{0})}^{d}$ and the matrix G is negative-definite matrix. We emphasize that $\nabla_{x} u_{0, i}^{'}$ , $i = n, \dots, n + τ - 1$ , are defined uniquely, although $u_{0, i}^{'}$ are defined up to a constant.

Now, assume that $m = - 1 / 2$ . We obtain that the correction term $u_{1}^{'}$ is a solution to the problem (4.15)–(4.16) and the term $u_{0}^{″}$ has order $ε^{2}$ and it satisfies (4.10) combined with compatibility condition (4.11). Following the same arguments as before, we find that the correction term $λ^{'}$ is given by $\begin{array}{l} (4.20) & λ^{'} = - {(\partial_{ν_{0}} u_{1}^{'}, u_{1}^{0})}_{Γ_{0}} + {(\partial_{ν_{0}} u_{1}^{0}, u_{1}^{'})}_{Γ_{0}} = - λ^{0} c_{0}^{2} | Ω_{0} | - {‖ \nabla_{x} u_{0}^{'} ‖}_{L^{2} (Ω_{0})}^{2}, \end{array}$ if $λ_{n}^{0}$ is a simple eigenvalue, while $λ^{'}$ is $\begin{matrix} (4.21) & λ_{j}^{'} b_{p}^{j} = - \sum_{k = n}^{n + τ - 1} b_{k}^{j} {(\nabla_{x} u_{0, k}^{'}, \nabla_{x} u_{0, p}^{'})}_{Ω_{0}} - λ_{n}^{0} u_{0, j}^{0} u_{0, p}^{0} | Ω_{0} |, for p = n, \dots, n + τ - 1, \end{matrix}$ if $λ_{n}^{0}$ is a τ-multiple eigenvalue.

As far as the justification procedure is concerned, the estimate (2.4) of the Theorem 2.1 for $m < 0$ is valid with $α = 0$ , $β = 1$ , $γ = min {1 - m, 2}$ , $λ_{n}^{0}$ being an eigenvalue of the problem (4.5)–(4.6) and $λ_{n}^{'}$ the correction term in (4.1), given by formulas (4.12), (4.18) and (4.20) in the case of a simple eigenvalue and by formulas (4.14), (4.19) and (4.21) for a multiple ones.

5. Asymptotic expansion for $m = 0$

We predict the following ansätze $\begin{array}{l} (5.1) & λ^{ε} = λ^{0} + ε λ^{'} + \dots, \\ (5.2) & u_{0}^{ε} (x) = u_{0}^{0} (x) + ε u_{0}^{'} (x) + \dots, x \in Ω_{0}, \\ (5.3) & u_{1}^{ε} (x) = u_{1}^{0} (x) + ε u_{1}^{'} (x) + \dots, x \in Ω_{1}, \end{array}$ where the dots denote the lower-order terms. The formulas (5.1)–(5.2) mean that the eigenpair $(λ^{ε}, {u_{0}^{ε}, u_{1}^{ε}})$ is expected to depend on the parameter ε continuously.

The leading term $u_{0}^{0}$ satisfies the problem (3.5). The correction term $u_{0}^{'}$ is a solution to problem $\begin{array}{l} (5.4) & - Δ_{x} u_{0}^{'} (x) = λ^{0} u_{0}^{0} (x), x \in Ω_{0}, \partial_{ν_{0}} u_{0}^{'} (x) = \partial_{ν_{0}} u_{1}^{0} (x), x \in Γ_{0}, \end{array}$ whose compatibility condition reads as $\begin{matrix} (5.5) & c_{0} = \frac{1}{λ^{0} | Ω_{0} |} \int_{Γ_{0}} \partial_{ν_{0}} u_{1}^{0} d s_{x} . \end{matrix}$

The leading terms $λ^{0}$ and $u_{1}^{0}$ in (5.1) and (5.3) solve the problem (4.5)–(4.6) along with the integral condition (5.5). Therefore, the variational formulation reads as $\begin{matrix} (5.6) & {(\nabla_{x} u_{1}^{0}, \nabla_{x} φ)}_{Ω_{1}} = λ^{0} {(u_{1}^{0}, φ)}_{Ω_{0}} + λ^{0} | Ω_{0} | {\overline{u}}_{1}^{0} \overline{φ}, \forall φ \in H_{∙}^{1} (Ω_{1}, Γ_{0}) . \end{matrix}$ Here $\overline{u}$ denotes the constant trace of function $u \in H^{1} (Ω_{1})$ on the boundary $Γ_{0}$ . Problem (5.6) admits the discrete spectrum given by the monotone unbounded sequence of eigenvalues (3.10) and the corresponding eigenfunctions $u_{1, n}^{0}$ are subject to the orthonormalization conditions $\begin{matrix} (5.7) & {(u_{1, i}^{0}, u_{1, j}^{0})}_{Ω_{1}} + | Ω_{0} | {\overline{u}}_{1, i}^{0} {\overline{u}}_{1, j}^{0} = δ_{i, j}, for i, j \in N . \end{matrix}$

As a solution to the problem (5.4), $u_{0}^{'}$ is a unique up to an additive constant, so that we assume that $u_{0}^{'} (x) : = {\tilde{u}}_{0}^{'} + {\hat{u}}_{0}^{'} (x)$ , where ${\tilde{u}}_{0}^{'}$ is a constant and ${\hat{u}}_{0}^{'}$ is a function in $H^{1} (Ω_{0})$ such that $\begin{matrix} (5.8) & \int_{Ω_{0}} {\hat{u}}_{0}^{'} (x) d x = 0 . \end{matrix}$ Then, the correction term $u_{1}^{'}$ satisfies the problem $\begin{array}{l} (5.9) & - Δ_{x} u_{1}^{'} (x) - λ^{0} u_{1}^{'} (x) = λ^{'} u_{1}^{0} (x), x \in Ω_{1}, \\ (5.10) & \partial_{ν_{1}} u_{1}^{'} (x) = 0, x \in Γ_{1}, u_{1}^{'} (x) = {\tilde{u}}_{0}^{'} + {\hat{u}}_{0}^{'} (x), x \in Γ_{0} . \end{array}$ The correction term $λ^{'}$ is determined by the compatibility condition to the problem (5.9)–(5.10). Hence, if $λ_{n}^{0}$ is a simple eigenvalue of problem (4.5)–(4.6) and due to (5.8), we have $\begin{array}{l} λ_{n}^{'} \int_{Ω_{1}} {| u_{1, n}^{0} (x) |}^{2} d x & = {(\partial_{ν_{0}} u_{1, n}^{0}, u_{1, n}^{'})}_{Γ_{0}} - {(\partial_{ν_{0}} u_{1, n}^{'}, u_{1, n}^{0})}_{Γ_{0}} \\ (5.11) & = {\tilde{u}}_{0, n}^{'} \int_{Γ_{0}} \partial_{ν_{0}} u_{1, n}^{0} d s_{x} + \int_{Γ_{0}} \partial_{ν_{0}} u_{1, n}^{0} {\hat{u}}_{0, n}^{'} d s_{x} - {\overline{u}}_{1, n}^{0} \int_{Γ_{0}} \partial_{ν_{0}} u_{1, n}^{'} d s_{x} \end{array}$ In order to compute explicitly $λ_{n}^{'}$ , we look for more terms in the asymptotic expansions, so that we assume that $\begin{array}{l} λ^{ε} = λ^{0} + ε λ^{'} + ε^{2} λ^{″} + \dots, \\ u_{0}^{ε} (x) = u_{0}^{0} (x) + ε u_{0}^{'} (x) + ε^{2} u_{0}^{″} (x) + \dots, x \in Ω_{0}, \\ u_{1}^{ε} (x) = u_{1}^{0} (x) + ε u_{1}^{'} (x) + ε^{2} u_{1}^{″} (x) + \dots, x \in Ω_{1} . \end{array}$ The problem satisfied by $u_{0, n}^{″}$ is given by $\begin{array}{l} - Δ_{x} u_{0, n}^{″} (x) = λ_{n}^{0} u_{0, n}^{'} (x) + λ_{n}^{'} u_{0, n}^{0}, x \in Ω_{0}, \\ \partial_{ν_{0}} u_{0, n}^{″} (x) = \partial_{ν_{0}} u_{1, n}^{'} (x), x \in Γ_{0} . \end{array}$ The compatibility condition is $\begin{matrix} (5.12) & \int_{Γ_{0}} \partial_{ν_{0}} u_{1, n}^{'} d s_{x} = λ_{n}^{0} | Ω_{0} | {\tilde{u}}_{0, n}^{'} + λ_{n}^{'} | Ω_{0} | u_{0, n}^{0}, \end{matrix}$ since $u_{0, n}^{'} (x) = {\tilde{u}}_{0, n}^{'} + {\hat{u}}_{0, n}^{0} (x)$ and ${\hat{u}}_{0, n}^{0}$ satisfies (5.8). In view of (5.5) and (5.12), (5.11) turns into $\begin{array}{l} λ_{n}^{'} (\int_{Ω_{1}} {| u_{1, n}^{0} (x) |}^{2} d x + {({\overline{u}}_{1, n}^{0})}^{2} | Ω_{0} |) = \int_{Γ_{0}} \partial_{ν_{0}} u_{1, n}^{0} {\hat{u}}_{0, n}^{'} d s_{x} . \end{array}$ Thanks to the orthonomalization conditions (5.7), we conclude that $\begin{array}{l} λ_{n}^{'} = \int_{Γ_{0}} \partial_{ν_{0}} u_{1, n}^{0} {\hat{u}}_{0, n}^{'} d s_{x} . \end{array}$

If $λ_{n}^{0}$ has multiplicity $τ ⩾ 2$ , the leading term of $u_{1, n}^{ε}, \dots, u_{1, n + τ - 1}^{ε}$ are predicted in the form of linear combinations $\begin{matrix} U_{1, j}^{0} (x) : = a_{n}^{j} u_{1, n}^{0} (x) + \dots + a_{n + τ - 1}^{j} u_{1, n + τ - 1}^{0} (x), for j = n, \dots, n + τ - 1, \end{matrix}$ where $u_{1, n}^{0}, \dots, u_{1, n + τ - 1}^{0}$ are the eigenfunctions of the problem (4.5)–(4.6) which are subject to the orthonormalization conditions (5.7). In addition, we assume that the coefficients $a_{n}^{j}, \dots, a_{n + τ - 1}^{j}$ satisfy the orthonormalization conditions (3.17). Therefore, the first-order corrector $U_{1, j}^{'}$ , for $j = n, \dots, n + τ - 1$ , satisfies the problem $\begin{array}{l} (5.13) & - Δ_{x} U_{1, j}^{'} (x) - λ_{n}^{0} U_{1, j}^{'} (x) = λ_{j}^{'} U_{1, j}^{0} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} U_{1, j}^{'} (x) = 0, x \in Γ_{1}, \\ (5.14) & U_{1, j}^{'} (x) = \sum_{k = n}^{n + τ - 1} a_{k}^{j} ({\tilde{u}}_{0, k}^{'} + {\hat{u}}_{0, k}^{'} (x)), x \in Γ_{0}, \end{array}$ where ${\tilde{u}}_{0, n}^{'}, \dots, {\tilde{u}}_{0, n + τ - 1}^{'}$ are constants and ${\hat{u}}_{0, n}^{'}, \dots, {\hat{u}}_{0, n + τ - 1}^{'}$ are functions in $H^{1} (Ω_{0})$ satisfying (5.8). Then, the compatibility condition to the problem (5.13)–(5.14) reads as, for $j = n, \dots, n + τ - 1$ , $\begin{array}{rcl} λ_{j}^{'} {(U_{1, j}^{0}, u_{1, q}^{0})}_{Ω_{1}} & = & \sum_{k = n}^{n + τ - 1} a_{k}^{j} ({\tilde{u}}_{0, k}^{'} \int_{Γ_{0}} \partial_{ν_{0}} u_{1, q}^{0} d s_{x} + \int_{Γ_{0}} \partial_{ν_{0}} u_{1, q}^{0} {\hat{u}}_{0, k}^{'} d s_{x}) \\ (5.15) & - {\overline{u}}_{1, q}^{0} \int_{Γ_{0}} \partial_{ν_{0}} U_{1, j}^{'} d s_{x}, \end{array}$ for $q = n, \dots, n + τ - 1$ . As in the case of $λ_{n}^{0}$ simple eigenvalue, we find that the term $u_{0, j}^{″}$ of order $ε^{2}$ in the asymptotic expansion of $u_{0}^{ε}$ satisfies the problem $\begin{array}{l} - Δ_{x} u_{0, j}^{″} (x) = λ_{n}^{0} \sum_{k = n}^{n + τ - 1} a_{k}^{j} ({\tilde{u}}_{0, k}^{'} + {\hat{u}}_{0, k}^{'} (x)) + λ_{j}^{'} \sum_{k = n}^{n + τ - 1} a_{k}^{j} u_{0, k}^{0}, x \in Ω_{0}, \\ \partial_{ν_{0}} u_{0, j}^{″} (x) = \partial_{ν_{0}} U_{1, j}^{'} (x), x \in Γ_{0}, \end{array}$ where the compatibility condition is given by $\begin{matrix} (5.16) & \int_{Γ_{0}} \partial_{ν_{0}} U_{1, j}^{'} d s_{x} = λ_{n}^{0} | Ω_{0} | \sum_{k = n}^{n + τ - 1} a_{k}^{j} {\tilde{u}}_{0, k}^{'} + λ_{j}^{'} | Ω_{0} | \sum_{k = n}^{n + τ - 1} a_{k}^{j} u_{0, k}^{0} . \end{matrix}$ Hence, (5.15) becomes $\begin{array}{l} λ_{j}^{'} {(U_{1, j}^{0}, u_{1, q}^{0})}_{Ω_{1}} = & \sum_{k = n}^{n + τ - 1} a_{k}^{j} \int_{Γ_{0}} {\hat{u}}_{0, k}^{'} \partial_{ν_{0}} u_{1, q}^{0} d s_{x} \\ - λ_{j}^{'} {\overline{u}}_{1, q}^{0} | Ω_{0} | \sum_{k = n}^{n + τ - 1} a_{k}^{j} {\overline{u}}_{1, k}^{0}, for q = n, \dots, n + τ - 1, \end{array}$ which implies that $\begin{matrix} λ_{j}^{'} (\sum_{k = n}^{n + τ - 1} a_{k}^{j} [{(u_{1, k}^{0}, u_{1, q}^{0})}_{Ω_{1}} + {\overline{u}}_{1, q}^{0} {\overline{u}}_{1, k}^{0} | Ω_{0} |]) = \sum_{k = n}^{n + τ - 1} a_{k}^{j} \int_{Γ_{0}} {\hat{u}}_{0, k}^{'} \partial_{ν_{0}} u_{1, q}^{0} d s_{x}, \end{matrix}$ for $q = n, \dots, n + τ - 1$ . In view of the orthonormalization conditions (5.7), we conclude that $\begin{matrix} λ_{j}^{'} a_{q}^{j} = \sum_{k = n}^{n + τ - 1} a_{k}^{j} \int_{Γ_{0}} {\hat{u}}_{0, k}^{'} \partial_{ν_{0}} u_{1, q}^{0} d s_{x}, for q = n, \dots, n + τ - 1 . \end{matrix}$ Hence, the τ first-order correction terms $λ_{n}^{'}, \dots, λ_{n + τ - 1}^{'}$ are the eigenvalues of the $(τ \times τ)$ real-valued matrix M whose entries are defined by $\begin{matrix} M_{q, k} : = \int_{Γ_{0}} \partial_{ν_{0}} u_{1, q}^{0} {\hat{u}}_{0, k}^{'} d s_{x}, for q, k = n, \dots, n + τ - 1 . \end{matrix}$ The claim of Theorem 2.1 is still true and the estimate (2.4) becomes $\begin{matrix} | λ^{ε} - λ_{n}^{0} - ε λ_{n}^{'} | ⩽ C_{N} ε^{3 / 2} . \end{matrix}$

6. Asymptotic expansion for $m = 1 / 2$

This case is discussed in more abstract setting in the textbook [23, Chapter VII], but for the convenience of the reader an independent proof is presented for the problem under consideration. The Helmholtz equation (1.2) gets rid of the small parameter ε $\begin{array}{l} - Δ_{x} u_{0}^{ε} (x) & = λ^{ε} u_{0}^{ε} (x), x \in Ω_{0} . \end{array}$ We perform the replacement (3.1), i.e. $v_{0}^{ϵ} (x) = ε^{- 1 / 2} u_{0}^{ε} (x)$ and $v_{1}^{ε} (x) = u_{1}^{ε} (x)$ . The asymptotics of eigenpairs $(λ^{ε}, {u_{0}^{ε}, u_{1}^{ε}})$ take the form $\begin{array}{l} λ^{ε} = λ^{0} + ε^{1 / 2} λ^{'} + \dots, v_{1}^{ϵ} (x) = v_{1}^{0} (x) + ε^{1 / 2} v_{1}^{'} (x) + \dots, x \in Ω_{1}, \\ v_{0}^{ϵ} (x) = v_{0}^{0} (x) + ε^{1 / 2} v_{0}^{'} (x) + \dots, x \in Ω_{0}, \end{array}$ where the dots denote the lower-order terms. The essential difference with respect to the other cases is the presence of two spectral limit problems. In fact, the leading term $v_{0}^{0}$ is determined from the problem $\begin{array}{l} (6.1) & - Δ_{x} v_{0}^{0} (x) = λ^{0} v_{0}^{0} (x), x \in Ω_{0}, \partial_{ν_{0}} v_{0}^{0} (x) = 0, x \in Γ_{0} . \end{array}$ The leading term $v_{1}^{0}$ solves the problem $\begin{array}{l} (6.2) & - Δ_{x} v_{1}^{0} (x) = λ^{0} v_{1}^{0} (x), x \in Ω_{1}, \\ (6.3) & \partial_{ν_{1}} v_{1}^{0} (x) = 0, x \in Γ_{1}, v_{1}^{0} (x) = 0, x \in Γ_{0} . \end{array}$ The problem for the correction terms $v_{0}^{'}$ is $\begin{array}{l} - Δ_{x} v_{0}^{'} (x) - λ^{0} v_{0}^{'} (x) = λ^{'} v_{0}^{0} (x), x \in Ω_{0}, \partial_{ν_{0}} v_{0}^{'} (x) = \partial_{ν_{0}} v_{1}^{0} (x), x \in Γ_{0} . \end{array}$ The correction term $v_{1}^{'}$ is determined by the problem (3.12)–(3.13).

Owing to the two limit problems, the procedure made for the convergence theorem must be slightly modified. We explain it briefly. According to the convergence (3.26) of eigenvalues $λ_{n}^{ε}$ , the weak formulation (2.1) and the normalization condition (2.3) with $m = 1 / 2$ , we deduce $\begin{matrix} {‖ \nabla_{x} v_{1, n}^{ε} ‖}_{L^{2} (Ω_{1})}^{2} + ε^{- 1} {‖ \nabla_{x} v_{0, n}^{ε} ‖}_{L^{2} (Ω_{0})}^{2} ⩽ C_{n} . \end{matrix}$ As in Section 3.1.1, $v_{0, n}^{ε}$ converges to zero strongly in $H^{1} (Ω_{0})$ and hence in $L^{2} (Ω_{0})$ , while $v_{1, n}^{ε}$ converges to some $v_{1, \bar{n}}^{0}$ weakly in $H^{1} (Ω_{1})$ and strongly in $L^{2} (Ω_{1})$ . If $v_{1, \bar{n}}^{0} \neq 0$ and $λ_{\bar{n}}^{0}$ belongs to the spectrum of the problem (6.2)–(6.3), the continuity of trace operator ensures that $v_{0, n}^{ε}$ converges to 0 in $L^{2} (Γ_{0})$ . Then the boundary condition (1.4) yields the strong convergence of $v_{1, n}^{ε}$ to 0 in $L^{2} (Γ_{0})$ , i.e. $v_{1, \bar{n}}^{0} \in H_{0}^{1} (Ω_{1}, Γ_{0})$ . Using the same arguments as in Section 3.1.1 combined with the fact that $v_{1, \bar{n}}^{0} \neq 0$ in $Ω_{1}$ and $v_{1, \bar{n}}^{0} = 0$ on $Γ_{0}$ , we deduce that the leading terms $λ_{\bar{n}}^{0}$ and $v_{1, \bar{n}}^{0}$ are characterized as the eigenpairs of the spectral problem (6.2)–(6.3).

Assume, now, that $v_{1, \bar{n}}^{0} = 0$ and $λ_{\bar{n}}^{0}$ does not belong to the spectrum of the problem (6.2)–(6.3). The previous arguments fail and we introduce the new normalization condition $\begin{matrix} (6.4) & {‖ v_{1, n}^{ε} ‖}_{L^{2} (Ω_{1})}^{2} + ε^{- 1} {‖ v_{0, n}^{ε} ‖}_{L^{2} (Ω_{0})}^{2} = ε^{- 1} . \end{matrix}$ The weak formulation (2.1) implies the bound $\begin{matrix} (6.5) & {‖ \nabla_{x} v_{1, n}^{ε} ‖}_{L^{2} (Ω_{1})}^{2} + ε^{- 1} {‖ \nabla_{x} v_{0, n}^{ε} ‖}_{L^{2} (Ω_{0})}^{2} ⩽ C_{n} ε^{- 1} . \end{matrix}$ Multiplying the inequalities (6.4) and (6.5) by ε, the quantities $‖ v_{0, n}^{ε} ‖_{H^{1} (Ω_{0})}$ and $‖ \nabla_{x} v_{0, n}^{ε} ‖_{H^{1} (Ω_{0})}$ are bounded and then $v_{0, n}^{ε}$ converges weakly in $H^{1} (Ω_{0})$ and strongly to $L^{2} (Ω_{0})$ to some function $v_{0, \bar{n}}^{0}$ . Moreover, the trace of $v_{0, n}^{ε}$ converges to the trace of $v_{0, \bar{n}}^{0}$ in $L^{2} (Γ_{0})$ . Finally, the limit passage as $ε \to 0$ in the weak formulation (2.1) leads to characterize $v_{0, \bar{n}}^{0}$ as the eigenfunction with associated eigenvalue $λ_{\bar{n}}^{0}$ of the problem (6.1) and the eigenfunctions $v_{0, \bar{n}}^{0}$ are normalized in $L^{2} (Ω_{0})$ . Indeed, bearing in mind that $λ_{\bar{n}}^{0}$ does not belong to the spectrum of the problem (6.2)–(6.3) and the convergence (3.26), for small $ε > 0$ $λ_{n}^{ε}$ is not an eigenvalue of the problem $\begin{array}{l} - Δ_{x} v_{1, n}^{ε} (x) = λ_{n}^{ε} v_{1, n}^{ε} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} v_{1, n}^{ε} (x) = 0, x \in Γ_{1}, v_{1, n}^{ε} (x) = 0, x \in Γ_{0} . \end{array}$ As a consequence, we have $\begin{matrix} (6.6) & {‖ v_{1, n}^{ε} ‖}_{H^{1} (Ω_{1})} ⩽ c {‖ v_{0, n}^{ε} ‖}_{H^{1 / 2} (Γ_{0})} ⩽ c {‖ v_{0, n}^{ε} ‖}_{H^{1} (Ω_{0})} ⩽ c . \end{matrix}$ The inequalities (6.4) and (6.6) lead to check the normalization condition of the eigenfunction $v_{0, \bar{n}}^{0}$ . The Theorem 2.1 is still valid with the estimate $\begin{matrix} | λ^{ε} - λ_{n}^{0} - ε^{1 / 2} λ_{n}^{'} | ⩽ C_{N} ε . \end{matrix}$

7. Asymptotic expansion for $m > 1 / 2$

We postulate the asymptotic expansions for the eigenvalue $λ^{ε}$ $\begin{matrix} (7.1) & λ^{ε} = ε^{2 m - 1} λ^{0} + ε^{2 m} λ^{'} + \dots . \end{matrix}$ For the corresponding eigenfunctions ${u_{0}^{ε}, u_{1}^{ε}}$ we consider an asymptotic expansion of the form $\begin{array}{l} (7.2) & u_{0}^{ε} = u_{0}^{0} + ε u_{0}^{'} + \dots, x \in Ω_{0}, \\ (7.3) & u_{1}^{ε} = u_{1}^{0} + ε^{min {1, 2 m - 1}} u_{1}^{'} + \dots, x \in Ω_{1} . \end{array}$ Here the dots denote the lower-order terms. Using the same procedure as in the other cases, we find that the leading terms $λ^{0}$ , $u_{0}^{0}$ in (7.1), (7.2) are characterized as the solution to the spectral problem $\begin{matrix} (7.4) & - Δ_{x} u_{0}^{0} (x) = λ^{0} u_{0}^{0} (x), x \in Ω_{0}, \partial_{ν_{0}} u_{0}^{0} (x) = 0, x \in Γ_{0} . \end{matrix}$ The problem (7.4) in the Sobolev space $H^{1} (Ω_{0})$ has a discrete spectrum $\begin{matrix} 0 = λ_{1}^{0} < λ_{2}^{0} ⩽ \dots ⩽ λ_{n}^{0} ⩽ \dots \to \infty \end{matrix}$ and the corresponding eigenfunctions $u_{0, n}^{0}$ are subject to the orthonormalization condition in $L^{2} (Ω_{0})$ . The leading term $u_{1}^{0}$ in (7.3) is defined as a unique solution of the problem $\begin{array}{l} - Δ_{x} u_{1}^{0} (x) = 0, x \in Ω_{1}, \\ \partial_{ν_{1}} u_{1}^{0} (x) = 0, x \in Γ_{1}, u_{1}^{0} (x) = u_{0}^{0} (x) x \in Γ_{0} . \end{array}$ If $min {1, 2 m - 1} = 2 m - 1$ , the problem for the correction term $u_{1}^{'}$ in (7.3) is $\begin{array}{l} - Δ_{x} u_{1}^{'} (x) = λ^{0} u_{1}^{0} (x), x \in Ω_{1}, \\ \partial_{ν_{1}} u_{1}^{'} (x) = 0, x \in Γ_{1}, u_{1}^{'} (x) = 0, x \in Γ_{0} . \end{array}$ If $min {1, 2 m - 1} = 1$ , the correction term $u_{1}^{'}$ is characterized as the solution to the problem $\begin{array}{l} - Δ_{x} u_{1}^{'} (x) = 0, x \in Ω_{1}, \\ \partial_{ν_{1}} u_{1}^{'} (x) = 0, x \in Γ_{1}, u_{1}^{'} (x) = u_{0}^{'} (x), x \in Γ_{0} . \end{array}$ We point out that when $m = 1$ , the problem satisfied by $u_{1}^{'}$ turns into $\begin{array}{l} - Δ_{x} u_{1}^{'} (x) = λ^{0} u_{1}^{0}, x \in Ω_{1}, \\ \partial_{ν_{1}} u_{1}^{'} (x) = 0, x \in Γ_{1}, u_{1}^{'} (x) = u_{0}^{'}, x \in Γ_{0} . \end{array}$ The correction term $u_{0}^{'}$ in the asymptotic expansion (7.2) is determined from the problem $\begin{array}{l} (7.5) & - Δ_{x} u_{0}^{'} (x) - λ^{0} u_{0}^{'} (x) = λ^{'} u_{0}^{0} (x), x \in Ω_{0}, \partial_{ν} u_{0}^{'} (x) = \partial_{ν} u_{1}^{0} (x), x \in Γ_{0} . \end{array}$ The compatibility condition in the problem (7.5) provides the correction term $λ^{'}$ . Indeed, if the eigenvalue $λ_{n}^{0}$ is simple then we get $\begin{matrix} (7.6) & λ^{'} = - {‖ \nabla_{x} u_{1}^{0} ‖}_{L^{2} (Ω_{1})}^{2} . \end{matrix}$ Assume now that the eigenvalue $λ_{n}^{0}$ has multiplicity $τ > 1$ , i.e. $λ_{n - 1}^{0} < λ_{n}^{0} = \dots = λ_{n + τ - 1}^{0} < λ_{n + τ}^{0}$ . The leading term in the expansions (7.2) are predicted in the form of linear combinations of the eigenfunctions $u_{0, n}^{0}, \dots, u_{0, n + τ - 1}^{0}$ $\begin{matrix} U_{0, j}^{0} (x) = a_{n}^{j} u_{0, n}^{0} (x) + \dots + a_{n + τ - 1}^{j} u_{0, n + τ - 1}^{0} (x), j = n, \dots, n + τ - 1 . \end{matrix}$ Therefore, $U_{0, j}^{'}$ is the solution to the problem $\begin{array}{l} - Δ_{x} U_{0, j}^{'} (x) - λ_{n}^{0} U_{0, j}^{'} (x) = λ_{j}^{'} U_{0, j}^{0} (x), x \in Ω_{0}, \\ \partial_{ν} U_{0, j}^{'} (x) = \sum_{k = n}^{n + τ - 1} a_{k}^{j} \partial_{ν_{0}} u_{1, k}^{0} (x), x \in Γ_{0} . \end{array}$ According to the Fredhom alternative, the τ compatibility conditions are $\begin{array}{l} λ_{j}^{'} {(U_{0, j}^{0}, u_{0, p}^{0})}_{Ω_{0}} & = {(\partial_{ν_{0}} U_{0, j}^{'}, u_{0, p}^{0})}_{Γ_{0}} \\ = \sum_{k = n}^{n + τ - 1} a_{n}^{j} {(\partial_{ν_{0}} u_{1, k}^{0}, u_{0, p}^{0})}_{Γ_{0}} \\ (7.7) & = - \sum_{k = n}^{n + τ - 1} a_{n}^{j} {(\nabla_{x} u_{1, k}^{0}, \nabla_{x} u_{0, p}^{0})}_{Ω_{1}}, j = n, \dots, n + τ - 1 \end{array}$ The previous relation can be written as the formula (4.19) with a different Gram matrix. In other words, the τ correction terms are the eigenvalues of the Gram matrix G whose entries are given by $\begin{matrix} G_{i . j} = - {(\nabla_{x} u_{1, i}^{0}, \nabla_{x} u_{1, j}^{0})}_{Ω_{1}}, i, j = n, \dots, n + τ - 1, \end{matrix}$ with $a_{n}^{j}$ being the corresponding eigenvectors. The estimate (2.4) of Theorem 2.1 holds with $α = 2 m - 1$ , $β = m$ , $γ = min {4 m - 1, 1 + m}$ for $m \in (1 / 2, 1)$ and $γ = 2 m + 1$ for $m ⩾ 1$ , $λ_{n}^{0}$ being the eigenvalue of the problem (7.4) and $λ_{n}^{'}$ being the correction term, given by formula (7.6) if $λ_{n}^{0}$ is a simple eigenvalue and (7.7) if $λ_{n}^{0}$ is a multiple one.

Fig. 2.

Kissing domains.

8. Kissing domains in

R^{2}

A distinguishing feature of the stiff Neumann problem is that all asymptotic forms derived and justified in previous sections, are preserved when $dist (Γ_{0}, Γ_{1}) \to 0$ , i.e. in the limit the core $Ω_{0}$ touches the exterior boundary $Γ_{1}$ , cf. Fig. 2, so that $Ω_{0}$ and Ω form the interior kiss of two domains. This peculiar conclusion is certainly based on the exterior Neumann condition (1.3) and the prevailing stiffness of the annulus $Ω_{1}$ . Changing particular details in problem’s statement may quit the above-mentioned limit passage: in Section 8.4 we will discuss a serious issue in the case when the Neumann condition (1.3) is replaced with the homogeneous Dirichlet one. In this way, in many cases the Dirichlet problem of the type (1.1)–(1.4) remains a fully open question.

The performed asymptotic analysis demonstrates that for $m ⩽ 1 / 2$ the limit problem in the cuspidal annulus reads $\begin{array}{l} (8.1) & - Δ_{x} u (x) = λ u (x), x \in Ω_{1} \\ (8.2) & \partial_{ν_{1}} u (x) = 0, x \in Γ_{1} ∖ O u (x) = g (x), x \in Γ_{0} ∖ O, \end{array}$ where $λ ⩾ 0$ and $g = 0$ or $g = const$ on the boundary $Γ_{0}$ . Denoting by $G \in H^{1} (R^{d} ∖ Ω_{0})$ an extension of g onto the exterior of $Ω_{0}$ , the variational formulation of the problem (8.1)–(8.2) reads, cf. [11]: find $u \in H^{1} (Ω_{1})$ such that $u - G \in H^{1} (Ω_{1}; Γ_{0})$ and the following integral identity is valid $\begin{matrix} (8.3) & {(\nabla_{x} u, \nabla_{x} v)}_{Ω_{1}} = λ {(u, v)}_{Ω_{1}} \forall v \in H, \end{matrix}$ with $H = H_{0}^{1} (Ω_{1}; Γ_{0})$ if $g = 0$ on $Γ_{0}$ or $H = H_{∙}^{1} (Ω_{1}, Γ_{0})$ if $g = const$ on $Γ_{0}$ Due to the Dirichlet condition on $Γ_{0}$ , the space $H_{0}^{1} (Ω_{1}; Γ_{0})$ is compactly embedded into $L^{2} (Ω_{1})$ .1

¹
This fact is true for $H^{1} (Ω_{1})$ as well, see, e.g. [15].

Hence, all necessary properties of the problem (8.1)–(8.2) are kept in the cuspidal domain

Ω_{1}

and these allow us to repeat with easily predictable modifications our calculations and argumentation in previous sections, to conclude the analog of the Theorem 2.1 in the case of the touching boundaries

Γ_{0}

and

Γ_{1}

. In the sequel we will describe the asymptotic behaviour as

x \to O

of solutions to the mixed boundary value problem (8.1)–(8.2) and a similar Dirichlet problem in

Ω_{1}

. However, to reduce cumbersome and long computations, we deal with the 2D case only while a needed modification for multi-dimensional cases can be performed in the same way described in the papers [16,18–20], where other types of singularities are dealt with.

8.1. Asymptotics of solutions at the cusp in Neumann case

We consider the spectral problem (8.1)–(8.2), where $g = c_{0}$ on $Γ_{0} ∖ O$ and $c_{0}$ is an arbitrary constant. Set $R_{0}$ , $R_{1}$ the radii of the disks $Ω_{0}$ , $Ω_{1}$ respectively such that $R_{0} < R_{1}$ . The boundaries $Γ_{0}$ and $Γ_{1}$ are described by $\begin{matrix} (8.4) & H_{i} (x_{1}) = \frac{| x_{1} |^{2}}{2 R_{i}} + O (| x_{1} |^{4}), i = 0, 1 . \end{matrix}$ The thickness is defined as $H (x_{1}) = H_{0} (x_{1}) - H_{1} (x_{1})$ . We write down the representation $\begin{matrix} (8.5) & u (x) = c_{0} + \dots, x \to O, \end{matrix}$ where the dots denote the lower-order terms. The distinguished asymptotic term on the right-hand side of (8.5) satisfies the boundary conditions (8.2) but generates the residual $\begin{matrix} λ c_{0} + \dots \end{matrix}$ in the differential equation (8.1). Then, we introduce a new term $U_{1} (x_{1}, η)$ in (8.5), involving the stretched coordinate $\begin{matrix} η = \frac{x_{2} - H_{1} (x_{1})}{H (x_{1})} \in (0, 1) . \end{matrix}$ Then the asymptotic (8.5) turns into $\begin{matrix} (8.6) & u (x) = c_{0} + U_{1} (x_{1}, η) + \dots, x \to O . \end{matrix}$ In order to rewrite (8.1) in the new variables $(x_{1}, η)$ , we evaluate $\begin{array}{l} (8.7) & \frac{\partial}{\partial x_{2}} = \frac{\partial η}{\partial x_{2}} \frac{\partial}{\partial η} = \frac{1}{H (x_{1})} \frac{\partial}{\partial η}, \frac{\partial^{2}}{\partial x_{2}^{2}} = \frac{1}{H {(x_{1})}^{2}} \frac{\partial^{2}}{\partial η^{2}}, \\ \frac{\partial}{\partial x_{1}} = \frac{\partial}{\partial x_{1}} - H {(x_{1})}^{- 1} (H_{1}^{'} (x_{1}) + η H^{'} (x_{1})) \frac{\partial}{\partial η}, \\ \frac{\partial^{2}}{\partial x_{2}^{2}} = {(\frac{\partial}{\partial x_{1}} - H {(x_{1})}^{- 1} (H_{1}^{'} (x_{1}) + η H^{'} (x_{1})) \frac{\partial}{\partial η})}^{2} \\ (8.8) & + (\frac{2 H^{'} (x_{1}) H_{1}^{'} (x_{1}) + 2 {(H^{'} (x_{1}))}^{2} η - H_{1}^{″} (x_{1}) H (x_{1}) - H^{″} (x_{1}) H (x_{1}) η}{H {(x_{1})}^{2}}) \frac{\partial}{\partial η}, \end{array}$ where $H^{'} (x_{1}) = \frac{d H (x_{1})}{d x_{1}}$ . Owing to (8.4), (8.7), (8.8), the Laplace operator $Δ_{(x_{1}, x_{2})}$ in the new variables $(x_{1}, η)$ is written as $\begin{array}{l} (8.9) & Δ_{(x_{1}, η)} & = \frac{1}{H^{p} {(x_{1})}^{2}} \frac{\partial^{2}}{\partial η^{2}} + \sum_{j = 1}^{\infty} L_{j} (x_{1}, η, \frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}), \end{array}$ where we replaced the thickness function $H (x_{1})$ with its principal part $\begin{matrix} H^{p} (x_{1}) = \frac{1}{2} (\frac{1}{R_{0}} - \frac{1}{R_{1}}) | x_{1} |^{2} . \end{matrix}$ The normal derivative on the lower boundary $Γ_{1}$ can be written as $\begin{array}{l} \partial_{ν_{1}} = & \frac{1}{{(1 + | H_{1}^{'} (x_{1}) |^{2})}^{1 / 2}} (\frac{\partial}{\partial x_{2}} - H_{1}^{'} (x_{1}) \frac{\partial}{\partial x_{1}}) \\ = & \frac{1}{{(1 + | H_{1}^{'} (x_{1}) |^{2})}^{1 / 2}} \\ (8.10) & \times (\frac{1}{H (x_{1})} \frac{\partial}{\partial η} - H_{1}^{'} (x_{1}) \frac{\partial}{\partial x_{1}} + H_{1}^{'} (x_{1}) H {(x_{1})}^{- 1} (H_{1}^{'} (x_{1}) + η H^{'} (x_{1})) \frac{\partial}{\partial η}) . \end{array}$ In view of (8.9) and (8.10), we insert the expansion ansatz (8.6) into the problem (8.1)–(8.2), obtaining the problem $\begin{array}{l} (8.11) & - \frac{1}{H^{p} {(x_{1})}^{2}} \frac{\partial^{2}}{\partial η^{2}} U_{1} (x_{1}, η) = λ c_{0}, \\ \frac{\partial}{\partial η} U_{1} {(x_{1}, η)}_{| η = 0} = 0, U_{1} {(x_{1}, η)}_{| η = 1} = 0 . \end{array}$ By a direct computation, the solution $U_{1}$ is given by $\begin{matrix} (8.12) & U_{1} (x_{1}, x_{2}) = - \frac{λ c_{0}}{2} [x_{2}^{2} - 2 H_{1}^{p} (x_{1}) (x_{2} + H_{0}^{p} (x_{1})) - H_{0}^{p} {(x_{1})}^{2}], \end{matrix}$ where $H_{i}^{p} (x_{1}) = | x_{1} |^{2} / (2 R_{i})$ denotes the principal part of $H_{i} (x_{1})$ , $i = 0, 1$ . Note that the first-order correction term $U_{1} (x_{1}, x_{2})$ is of order $| x_{1} |^{4}$ . Iterating this procedure, we are able to construct the formal infinite series of the eigenfunction u of the problem (8.1)–(8.2) $\begin{matrix} (8.13) & u (x) = c_{0} + \sum_{j = 1}^{\infty} U_{j} (x), \end{matrix}$ containing the already chosen main term (8.12). Keeping in mind the decomposition (8.9), (8.10) and replacing the eigenfunction u with its formal series (8.13) into the equation (8.1), we find that the term $U_{2}$ is solution of the problem $\begin{array}{l} - \frac{1}{H^{p} {(x_{1})}^{2}} \frac{\partial^{2}}{\partial η^{2}} U_{2} (x_{1}, η) = L_{1} (x_{1}, η, \frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}, λ) U_{1} (x_{1}, η), \\ \partial_{η} U_{2} {(x_{1}, η)}_{| η = 0} = N_{1} (\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}) U_{1} {(x_{1}, η)}_{| η = 0}, U_{2} {(x_{1}, η)}_{| η = 1} = 0, \end{array}$ where $\begin{array}{l} L_{1} (x_{1}, η, \frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}, λ) \\ = {(\frac{\partial}{\partial x_{1}} - H^{p} {(x_{1})}^{- 1} ({(H_{1}^{p})}^{'} (x_{1}) + η {(H^{p})}^{'} (x_{1})) \frac{\partial}{\partial η})}^{2}, \\ + (\frac{2 {(H^{p} (x_{1}))}^{'} {(H_{1}^{p} (x_{1}))}^{'} + 2 {(H^{' p} (x_{1}))}^{2} η}{H^{p} {(x_{1})}^{2}} \\ - \frac{{(H_{1}^{p})}^{″} (x_{1}) H^{p} (x_{1}) + {(H^{p} (x_{1}))}^{″} H^{p} (x_{1}) η}{H^{p} {(x_{1})}^{2}}) \frac{\partial}{\partial η} + \frac{| x_{1} |^{6}}{8 R^{4}} λ c_{0} \\ N_{1} (\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}) = {(H_{1}^{p})}^{'} (x_{1}) (\frac{\partial}{\partial x_{1}} + \frac{{(H_{1}^{p})}^{'} (x_{1})}{{(H^{p})}^{'} (x_{1})} \frac{\partial}{\partial η}) . \end{array}$ The other terms of the series (8.13) are determined by the problems $\begin{array}{l} - \frac{1}{H^{p} (x_{1})} \partial_{η} U_{j} (x_{1}, η) = L_{j - 1} (x_{1}, η, \frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}, λ) U_{j - 1} (x_{1}, η) + λ U_{j - 2} (x_{1}, η), j = 3, 4, \dots, \\ \partial_{η} U_{j} {(x_{1}, η)}_{| η = 0} = N_{j - 1} (0, \frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial η}) U_{j - 1} {(x_{1}, η)}_{| η = 0}, U_{j} {(x_{1}, η)}_{| η = 1} = 0 . \end{array}$ We point out that the terms $U_{j}$ , $j = 2, 3, \dots$ , of the series (8.13) are of order $| x_{1} |^{2 j + 2}$ .

8.2. Justification of asymptotics

Let χ be a smooth cut-off function such that $0 ⩽ χ (x_{1}) ⩽ 1$ and $\begin{matrix} χ (x_{1}) = 0, if | x_{1} | ⩾ R_{0}, χ (x_{1}) = 1 if | x_{1} | ⩽ \frac{R_{0}}{2} . \end{matrix}$ We set $\begin{matrix} (8.14) & u (x) = c_{0} + χ (x_{1}) U_{1} (x) + \tilde{u} (x), \end{matrix}$ with $\tilde{u} (x)$ being the remainder.

Theorem 8.1.
The solution u of the spectral problem ( 8.1 )–( 8.2 ) admits the asymptotic form ( 8.14 ). More specifically, there exists an exponent $N > 0$ such that the norm $\begin{matrix} {‖ ρ {(x)}^{- N} \tilde{u} ‖}_{H_{0}^{1} (Ω_{1}, Γ_{0})} < \infty \end{matrix}$ and the functions $ρ {(x)}^{- N} (u (x) - c_{0})$ and $ρ {(x)}^{- N} U_{1} (x_{1}, η)$ do not belong to the Sobolev space $H^{1}$ in a neighbourhood of the cusp $O$ .
Proof.
The remainder $\tilde{u}$ verifies the following equation $\begin{matrix} (8.15) & - Δ_{x} \tilde{u} (x) - λ \tilde{u} (x) = λ c_{0} + λ U_{1} (x) χ (x_{1}) + Δ_{x} (χ (x_{1}) U_{1} (x)), x \in Ω_{1} ∖ O, \end{matrix}$ along with homogeneous boundary conditions $\begin{matrix} \partial_{ν_{1}} \tilde{u} (x) = 0, x \in Γ_{1} ∖ O, \tilde{u} (x) = 0, x \in Γ_{0} ∖ O . \end{matrix}$ Multiplying (8.15) by an arbitrary test function $v \in H_{0}^{1} (Ω_{1}, Γ_{0})$ and integrating in $Ω_{1}$ , we find $\begin{array}{l} {(- Δ_{x} \tilde{u}, v)}_{Ω_{1}} - λ {(\tilde{u}, v)}_{Ω_{1}} \\ = {(Δ_{x} (U_{1} χ), v)}_{Ω_{1}} + λ {(c_{0}, v)}_{Ω_{1}} + λ {(U_{1} χ, v)}_{Ω_{1}} \\ (8.16) & = {(χ Δ_{x} U_{1}, v)}_{Ω_{1}} + {([Δ_{x}, χ] U_{1}, v)}_{Ω_{1}} + λ {(c_{0}, v)}_{Ω_{1}} + λ {(U_{1} χ, v)}_{Ω_{1}} . \end{array}$ The commutator $[Δ_{x}, χ]$ is $\begin{matrix} [Δ_{x}, χ] U = 2 \nabla_{x} U \cdot \nabla_{x} χ + U Δ_{x} χ . \end{matrix}$ Let ρ denote a smooth positive function on $Ω_{1}$ which coincides with the distance to the origin of the Cartesian coordinate system in a neighbourhood of the cuspidal point $O$ and introduce the weight function $\begin{matrix} T_{δ} (x) = \{\begin{matrix} δ^{- N}, & if ρ (x) ⩽ δ, \\ ρ {(x)}^{- N}, & if δ < ρ (x) ⩽ R_{0} / 2, \\ {(R_{0} / 2)}^{- N}, & if ρ (x) > R_{0} / 2, \end{matrix} \end{matrix}$ where the parameter $δ > 0$ is small and will be sent to 0. Later on, we will impose some constraints on the exponent N. We point out that the derivative of $T_{δ}$ vanishes for $ρ (x) ⩽ δ$ , $ρ (x) > R_{0} / 2$ and satisfies the inequality $\begin{matrix} (8.17) & | \nabla_{x} T_{δ} (x) | ⩽ C T_{δ} (x) ρ {(x)}^{- 1}, δ < ρ (x) ⩽ R_{0} . \end{matrix}$ Since $\tilde{u} \in H_{0}^{1} (Ω_{1}, Γ_{0})$ , we choose as a test function $V = T_{δ} \tilde{v} \in H_{0}^{1} (Ω_{1}, Γ_{0})$ , with $\tilde{v} = T_{δ} \tilde{u}$ . After algebraic transformations, the left-hand side of (8.16) can be written as $\begin{array}{l} - {(Δ_{x} \tilde{u}, V)}_{Ω_{1}} - λ {(\tilde{u}, V)}_{Ω_{1}} = & {(\nabla_{x} \tilde{u}, \nabla_{x} V)}_{Ω_{1}} - λ {(\tilde{v}, \tilde{v})}_{Ω_{1}} \\ = & {(\nabla_{x} \tilde{u}, \tilde{v} \nabla_{x} T_{δ})}_{Ω_{1}} + {(\nabla_{x} \tilde{u}, T_{δ} \nabla_{x} \tilde{v})}_{Ω_{1}} - λ {(\tilde{v}, \tilde{v})}_{Ω_{1}} \\ = & {(T_{δ} \nabla_{x} \tilde{u}, \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ})}_{Ω_{1}} + {(T_{δ} \nabla_{x} \tilde{u}, \nabla_{x} \tilde{v})}_{Ω_{1}} - λ {(\tilde{v}, \tilde{v})}_{Ω_{1}} \\ = & {(\nabla_{x} \tilde{v}, \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ})}_{Ω_{1}} - {(\tilde{u} \nabla_{x} T_{δ}, \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ})}_{Ω_{1}} + {(\nabla_{x} \tilde{v}, \nabla_{x} \tilde{v})}_{Ω_{1}} \\ - {(\tilde{u} \nabla_{x} T_{δ}, \nabla_{x} \tilde{v})}_{Ω_{1}} - λ {(\tilde{v}, \tilde{v})}_{Ω_{1}} \\ = & {(\nabla_{x} \tilde{v}, \nabla_{x} \tilde{v})}_{Ω_{1}} - {(\tilde{u} \nabla_{x} T_{δ}, \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ})}_{Ω_{1}} - λ {(\tilde{v}, \tilde{v})}_{Ω_{1}} \\ (8.18) & = & ‖ \nabla_{x} \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} - {‖ \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ} ‖}_{L^{2} (Ω_{1})}^{2} - λ ‖ \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} . \end{array}$ From formulas (8.16) and (8.18), we find that $\begin{array}{l} ‖ \nabla_{x} \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} = & {(χ Δ_{x} U_{1}, V)}_{Ω_{1}} + {([Δ_{x}, χ] U_{1}, V)}_{Ω_{1}} + λ {(c_{0}, V)}_{Ω_{1}} + λ {(U_{1} χ, V)}_{Ω_{1}} \\ (8.19) & + {‖ \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ} ‖}_{L^{2} (Ω_{1})}^{2} + λ ‖ \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} . \end{array}$ We estimate each terms in the previous equality. Bearing in mind that the correction term $U_{1}$ is the solution of (8.11), we obtain $\begin{array}{l} {(χ Δ_{x} U_{1}, V)}_{Ω_{1}} = {(χ \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1}, V)}_{Ω_{1}} + {(χ \frac{\partial^{2}}{\partial x_{2}^{2}} U_{1}, V)}_{Ω_{1}} = {(χ \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1}, V)}_{Ω_{1}} - {(χ λ c_{0}, V)}_{Ω_{1}} . \end{array}$ Therefore, $\begin{array}{l} | λ {(c_{0}, V)}_{Ω_{1}} - {(χ λ c_{0}, V)}_{Ω_{1}} | ⩽ | λ {(c_{0}, V)}_{Ω_{1} \cap {x : ρ (x) ⩾ R_{0} / 2}} | ⩽ λ c_{0} {(R_{0} / 2)}^{- 2 N} ‖ \tilde{u} ‖_{L^{2} (Ω_{1})}^{2} < \infty . \end{array}$ Moreover, from Poincarè’s inequality $\begin{matrix} (8.20) & {‖ H {(R_{0})}^{- 1} \tilde{v} (x) ‖}_{L^{2} (Ω_{1})} ⩽ C {‖ \nabla_{x} \tilde{v} (x) ‖}_{L^{2} (Ω_{1})}, \end{matrix}$ we find $\begin{array}{l} | {(χ \frac{\partial^{2}}{\partial x^{2}} U_{1}, V)}_{Ω_{1}} | & = | {(χ T_{δ} \frac{\partial^{2}}{\partial x^{2}} U_{1}, \tilde{v})}_{Ω_{1}} | \\ = | {(χ T_{δ} H (x_{1}) \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1}, H {(x_{1})}^{- 1} \tilde{v})}_{Ω_{1}} | \\ ⩽ {‖ χ T_{δ} H (x_{1}) \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1} ‖}_{L^{2} (Ω_{1})} {‖ H {(x_{1})}^{- 1} \tilde{v} ‖}_{L^{2} (Ω_{1})} \\ ⩽ C {‖ χ T_{δ} H (x_{1}) \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1} ‖}_{L^{2} (Ω_{1})} ‖ \nabla_{x} \tilde{v} ‖_{L^{2} (Ω_{1})} . \end{array}$ Taking into account that $H (x_{1}) = O (| x_{1} |^{2})$ and $\frac{\partial^{2}}{\partial x_{1}^{2}} U_{1} = O (| x_{1} |^{2})$ , the norm $\begin{array}{l} {‖ χ T_{δ} H (x_{1}) \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1} ‖}_{L^{2} (Ω_{1})} ⩽ & {‖ ρ^{- N} H (x_{1}) \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1} ‖}_{L^{2} (Ω_{1} \cap {x : ρ (x) ⩽ R_{0} / 2})} \\ + {(R_{0} / 2)}^{- 2 N} {‖ H (x_{1}) \frac{\partial^{2}}{\partial x_{1}^{2}} U_{1} ‖}_{L^{2} (Ω_{1} \cap {x : ρ (x) > R_{0} / 2})} \end{array}$ is finite for $N < \frac{11}{5}$ . The term ${([Δ_{x}, χ] U_{1}, V)}_{Ω_{1}}$ involves the derivatives of the cut-off function χ. Then it does not vanish only if $R_{0} / 2 < ρ (x) < R_{0}$ and $\begin{array}{l} | {([Δ_{x}, χ] U_{1}, V)}_{Ω_{1}} | = & | {([Δ_{x}, χ] U_{1}, V)}_{{x \in Ω_{1} : R_{0} / 2 < ρ (x) < R_{0}}} | \\ = & | {(T_{δ} [Δ_{x}, χ] U_{1}, \tilde{v})}_{{x \in Ω_{1} : R_{0} / 2 < ρ (x) < R_{0}}} | \\ ⩽ & {‖ T_{δ} [Δ_{x}, χ] H (x_{1}) U_{1} ‖}_{L^{2} ({x \in Ω_{1} : R_{0} / 2 < ρ (x) < R_{0}})} \\ \times {‖ H^{- 1} (x_{1}) \tilde{v} ‖}_{L^{2} ({x \in Ω_{1} : R_{0} / 2 < ρ (x) < R_{0}})} \\ ⩽ & C {‖ T_{δ} [Δ_{x}, χ] H (x_{1}) U_{1} ‖}_{L^{2} ({x \in Ω_{1} : R_{0} / 2 < ρ (x) < R_{0}})} \\ \times ‖ \nabla_{x} \tilde{v} ‖_{L^{2} ({x \in Ω_{1} : R_{0} / 2 < ρ (x) < R_{0}})}, \end{array}$ which is finite for any value of N since $T_{δ} (x) = {(R_{0} / 2)}^{- N}$ if $ρ (x) > R_{0} / 2$ . According to the inequality (8.17), we have $\begin{matrix} {‖ \tilde{v} T_{δ}^{- 1} \nabla_{x} T_{δ} ‖}_{L^{2} (Ω_{1})}^{2} ⩽ C {‖ ρ^{- 1} \tilde{v} ‖}_{L^{2} (Ω_{1})}^{2} . \end{matrix}$ Choosing $R_{0}$ such that $λ ⩽ C H {(R_{0})}^{- 2}$ , from (8.20) we deduce $\begin{array}{l} λ ‖ \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} & ⩽ C H {(R_{0})}^{- 2} ‖ \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} ⩽ C ‖ \nabla_{x} \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} . \end{array}$ Finally, we have $\begin{array}{l} | {(U_{1} χ, V)}_{Ω_{1}} | & = | {(T_{δ} U_{1} χ, \tilde{v})}_{Ω_{1}} | \\ = | {(H (x_{1}) T_{δ} U_{1} χ, H {(x_{1})}^{- 1} \tilde{v})}_{Ω_{1}} | \\ ⩽ {‖ H (x_{1}) T_{δ} U_{1} χ ‖}_{L^{2} (Ω_{1})} {‖ H {(x_{1})}^{- 1} \tilde{v} ‖}_{L^{2} (Ω_{1})} \\ ⩽ C {‖ H (x_{1}) T_{δ} U_{1} χ ‖}_{L^{2} (Ω_{1})} ‖ \nabla_{x} \tilde{v} ‖_{L^{2} (Ω_{1})} . \end{array}$ The norm $\begin{array}{l} {‖ χ H (x_{1}) T_{δ} U_{1} ‖}_{L^{2} (Ω_{1})} ⩽ & {‖ ρ^{- N} H (x_{1}) U_{1} ‖}_{L^{2} (Ω_{1} \cap {x : ρ (x) ⩽ R_{0} / 2})} \\ + {(R_{0} / 2)}^{- 2 N} {‖ H (x_{1}) U_{1} ‖}_{L^{2} (Ω_{1} \cap {x : ρ (x) > R_{0} / 2})} \end{array}$ is finite if and only if $N < 15 / 2$ . Setting $N < 11 / 2$ , the relation (8.19) implies that $\begin{matrix} {‖ ρ^{- 1} \tilde{v} ‖}_{L^{2} (Ω_{1})}^{2} + ‖ \nabla_{x} \tilde{v} ‖_{L^{2} (Ω_{1})}^{2} ⩽ C < \infty . \end{matrix}$ Since $T_{δ}$ is monotone increasing as $δ \to 0$ , the limit of the last, bounded expression exists. □

Since terms in the formal series (8.13) are polynomials in x, we deduce the smoothness of the solution u to the problem (8.1)–(8.2). In virtue of estimate of Theorem 8.1, the estimate of the remainder in a weighted $H^{2}$ follows from standard local estimates near smooth parts of the boundary (cf. [20]).
8.3. The Dirichlet case

If we replace the boundary condition (8.2) on $Γ_{0}$ with a homogeneous Dirichlet one, i.e. $c_{0} = 0$ , then all eigenfunctions u of the problem $\begin{array}{l} (8.21) & - Δ_{x} u (x) = λ u (x), x \in Ω_{1}, \\ (8.22) & \partial_{ν_{1}} u (x) = 0, x \in Γ_{1} ∖ O, u (x) = 0, x \in Γ_{0} ∖ O, \end{array}$ decay exponentially as $x \to O$ .

Proposition 8.2.
The eigenfunction $u \in H_{0}^{1} (Ω_{1}, Γ_{0})$ of the problem ( 8.21 )–( 8.22 ) decays exponentially as $x \to O$ .
Proof.
Let $T_{δ}$ be the weight function defined by $\begin{matrix} T_{δ} (x) = \{\begin{matrix} e^{\frac{β}{δ}}, & | x_{1} | ⩽ δ, \\ e^{\frac{β}{| x_{1} |}}, & δ < | x_{1} | ⩽ R, \\ e^{\frac{β}{R}}, & | x_{1} | > R . \end{matrix} \end{matrix}$ Here, the parameter δ is small, positive and it will be sent to 0 and $β > 0$ . Note that $T_{δ}$ is a continuous function such that $\begin{matrix} | \nabla_{x} T_{δ} (x) | ⩽ β | x_{1} |^{- 2} T_{δ} (x), e^{\frac{β}{R}} ⩽ T_{δ} (x) ⩽ e^{\frac{β}{δ}} . \end{matrix}$ We insert into the integral identity (8.3) the test function $v = T_{δ} U \in H_{0}^{1} (Ω_{1}, Γ_{0})$ , with $U = T_{δ} u$ , obtaining $\begin{array}{l} λ {(u, v)}_{Ω_{1}} = & {(\nabla_{x} u, \nabla_{x} v)}_{Ω_{1}} \\ = & {(\nabla_{x} u, U \nabla_{x} T_{δ})}_{Ω_{1}} + {(\nabla_{x} u, T_{δ} \nabla_{x} U)}_{Ω_{1}} \\ = & {(T_{δ} \nabla_{x} u, T_{δ}^{- 1} U \nabla_{x} T_{δ})}_{Ω_{1}} + {(T_{δ} \nabla_{x} u, \nabla_{x} U)}_{Ω_{1}} \\ = & {(\nabla_{x} U, T_{δ}^{- 1} U \nabla_{x} T_{δ})}_{Ω_{1}} - {(u \nabla_{x} T_{δ}, T_{δ}^{- 1} U \nabla_{x} T_{δ})}_{Ω_{1}} \\ + {(\nabla_{x} U, \nabla_{x} U)}_{Ω_{1}} - {(u \nabla_{x} T_{δ}, \nabla_{x} U)}_{Ω_{1}} \\ = & {(\nabla_{x} U, \nabla_{x} U)}_{Ω_{1}} - {(T_{δ}^{- 1} U \nabla_{x} T_{δ}, T_{δ}^{- 1} U \nabla_{x} T_{δ})}_{Ω_{1}} . \end{array}$ Hence, $\begin{matrix} ‖ \nabla_{x} U ‖_{L^{2} (Ω_{1})}^{2} = λ ‖ U ‖_{L^{2} (Ω_{1})}^{2} + {‖ T_{δ}^{- 1} U \nabla_{x} T_{δ} ‖}_{L^{2} (Ω_{1})}^{2} . \end{matrix}$ Taking into account Poincarè’s inequality (8.20), we find $\begin{matrix} (c - β^{2}) {‖ | x_{1} |^{- 2} U ‖}_{L^{2} (Ω_{1})}^{2} ⩽ λ ‖ U ‖_{L^{2} (Ω_{1})}^{2} ⩽ λ e^{\frac{2 β}{δ}} ‖ u ‖_{L^{2} (Ω_{1})}^{2} < \infty . \end{matrix}$ In particular, choosing β such that $0 ⩽ β^{2} < c$ , we get $\begin{matrix} e^{- \frac{2 β}{δ}} \int_{Ω_{1}} | x_{1} |^{- 4} {| U (x) |}^{2} d x ⩽ c_{λ} < \infty . \end{matrix}$ It implies that both of the integrals $\begin{matrix} e^{- \frac{2 β}{δ}} \int_{Ω_{1} \cap {x : | x_{1} | ⩽ e^{- \frac{β}{2 δ}}}} | x_{1} |^{- 4} {| U (x) |}^{2} d x, e^{- \frac{2 β}{δ}} \int_{Ω_{1} \cap {x : | x_{1} | > e^{- \frac{β}{2 δ}}}} | x_{1} |^{- 4} {| U (x) |}^{2} d x, \end{matrix}$ are bounded for all $δ > 0$ . The first one gives $\begin{matrix} \int_{Ω_{1} \cap {x : | x_{1} | ⩽ e^{- \frac{β}{2 δ}}}} {| U (x) |}^{2} d x ⩽ e^{- \frac{2 β}{δ}} \int_{Ω_{1} \cap {x : | x_{1} | ⩽ e^{- \frac{β}{2 δ}}}} | x_{1} |^{- 4} {| U (x) |}^{2} d x < \infty \end{matrix}$ for all $δ > 0$ . Since $T_{δ}$ is monotone increase as $δ \to 0$ , we conclude that the eigenfunction u has an exponential decay in $L^{2}$ -norm in a neighbourhood of the cusp $O$ . □

The eigenfunctions u are thus smooth at any distance of $O$ and vanish at cusp point $O$ with all their derivatives due to the exponential decay. We conclude that also in this case the asymptotic anzätze for $(λ^{ε}, u^{ε})$ and the procedure given in the Sections 2–3 are correct.
8.4. Open questions

Due to the shape of the boundary $Γ_{1}$ , the solution of the problem (8.1)–(8.2) behaves in substantially different way from the solution of the problem $\begin{array}{l} (8.23) & - Δ_{x} u (x) = λ u (x), x \in Ω_{1}, \\ (8.24) & u (x) = 0, x \in Γ_{1} ∖ O, u (x) = c_{0}, x \in Γ_{0} ∖ O . \end{array}$ Here we have simply replaced the Neumann boundary condition (1.3) of the problem (8.1)–(8.2) with a homogeneous Dirichlet condition. Indeed an approximation of the solution u of the problem (8.23)–(8.24) is to be found in such a way that the boundary conditions are satisfied exactly while discrepancies in the equation (8.23) is reduced as much as possible. As a consequence, a solution u with the asymptotic $\begin{matrix} u (x) = c_{0} \frac{x_{2} - H_{1} (x_{1})}{H (x_{1})} + \dots, x \to O, \end{matrix}$ cannot belong to the Sobolev space $H^{1} (Ω_{1})$ . Indeed the integral $\begin{matrix} \int_{0}^{1 / 3} \int_{H_{1} (x_{1})}^{H_{0} (x_{1})} {| \frac{\partial}{\partial x_{2}} (c_{0} \frac{x_{2} - H_{1} (x_{1})}{H (x_{1})}) |}^{2} d x_{2} d x_{1} = 2 c_{0}^{2} \int_{0}^{1 / 3} \frac{1}{H {(x_{1})}^{2}} d x_{1} \end{matrix}$ is divergent because the integrand has nonadmissible singularity $O (| x_{1} |^{- 4})$ . The derivation of the ansatz for the eigenfunction u of the problem (8.23)–(8.24) is still an open problem.

Footnotes

Acknowledgements

The work of S.A. Nazarov has been supported by the Ministry of Science and Higher Education of Russian Federation within the framework of the Russian State Assignment under contract No. FFNF-2021-0006. The author S.A. Nazorv gratefully acknowledge the hospitality of the Department of Mathematical Sciences “G.L. Lagrange”, Dipartimento di Eccellenza 2018-2022, of Politecnico di Torino, and the support of GNAMPA-INdAM. The authors V. Chiadò Piat and L. D’Elia are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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The stiff Neumann problem: Asymptotic specialty and “kissing” domains

Abstract

Keywords

1. Introduction

3.1. Formal asymptotics in the case 0 < m < 1 / 2

3.1.1. Problem for v 0 0 and v 0 ′

3.1.2. Problem for v 1 0 and v 1 ′

3.1.3. Final remarks

3.2. Justification of asymptotics in the case m ∈ ( 0 , 1 / 2 )

3.2.1. Step 1: Convergence theorem

5. Asymptotic expansion for m = 0

6. Asymptotic expansion for m = 1 / 2

7. Asymptotic expansion for m > 1 / 2

1 This fact is true for H 1 ( Ω 1 ) as well, see, e.g. [15].

8.2. Justification of asymptotics

Footnotes

Acknowledgements

References

3.1. Formal asymptotics in the case $0 < m < 1 / 2$

3.1.1. Problem for $v_{0}^{0}$ and $v_{0}^{'}$

3.1.2. Problem for $v_{1}^{0}$ and $v_{1}^{'}$

3.2. Justification of asymptotics in the case $m \in (0, 1 / 2)$

5. Asymptotic expansion for $m = 0$

6. Asymptotic expansion for $m = 1 / 2$

7. Asymptotic expansion for $m > 1 / 2$

¹
This fact is true for $H^{1} (Ω_{1})$ as well, see, e.g. [15].